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=Introduction and Prerequisites=
 
=Introduction and Prerequisites=
  
−
Welcome to Multivariate Calculus!  This course will move quickly, and cover a lot of ground.  For this reason, the instructors personal email and phone number will be available to students.  We will begin with a discussion of functions of multiple variables, including differentiaion and integration, and move on to more complicated matters associated with these functions, such as optimization with constraints, surface integrals, and the basics of differential geometry.  It will cover the same material as a University-level Calculus 3 course, but we will begin with a review of concepts from Calculus 2 that the student should already be familiar with.
+
Welcome to Multivariable Calculus!  This course will move quickly, and cover a lot of ground.  For this reason, the instructors personal email and phone number will be available to students.  We will begin with a discussion of functions of multiple variables, including differentiation and integration, and move on to more complicated matters associated with these functions, such as optimization with constraints, surface integrals, and the basics of differential geometry.  It will cover the same material as a University-level Calculus 3 course, but we will begin with a review of concepts from Calculus 2 that the student should already be familiar with.
  
 
It is essential that the student have firm grasp of concepts from Calculus 1.  This includes, but is not limited to,
 
It is essential that the student have firm grasp of concepts from Calculus 1.  This includes, but is not limited to,
  
−
*Limits of functions of a single variable, including L'Hopitals rule,
+
*Limits of functions of a single variable, including [[L'Hopital's Rule|L'Hôpitals rule]],
−
*Differention of a function of a single variable,  
+
*Differentiation of a function of a single variable,  
 
*Integration of a function of a single variable, including integration by parts,
 
*Integration of a function of a single variable, including integration by parts,
 
*The fundamental theorem of calculus
 
*The fundamental theorem of calculus
 
*Analytic geometry of the plane, including arc length calculations
 
*Analytic geometry of the plane, including arc length calculations
 
*Infinite series, including the classic convergence tests and methods of calculation
 
*Infinite series, including the classic convergence tests and methods of calculation
 +
 +
The problems are absolutely essential to success in the course.  It is by doing exercises that the student gains an understanding of a mathematical concept, not by rote memorization of a definition.  Additionally, many peripheral concepts are discussed in the problems which are not touched upon, or only briefly discussed, in the main body of the lecture.  Students are encouraged to do all the review exercises and all the of the main exercises, but those students who find the review material unnecessary may instead choose to complete the challenging exercises.
  
 
=Course Outline=
 
=Course Outline=
Line 23: Line 43:
 
*'''Scalar Functions in Two and Three Dimensions'''
 
*'''Scalar Functions in Two and Three Dimensions'''
  
−
===Lecture 2===
+
===[[Calc3.2|Lecture 2]]===
 
*'''Vectors, and Vector Functions'''
 
*'''Vectors, and Vector Functions'''
  
 
*'''Coordinate Systems'''
 
*'''Coordinate Systems'''
  
−
*'''Limits of Multiariate Functions'''
+
*'''Limits of Multivariate Functions'''
  
−
*'''Elementary Vector Calculus'''
+
*'''Partial Derivatives'''
  
−
*'''Partial Derivatives'''
+
*'''The Jacobian'''
 +
 
 +
*'''Double and Triple Integrals'''
  
−
===Lecture 3===
+
===[[Calc3.3|Lecture 3]]===
 
*'''Directional Derivatives'''
 
*'''Directional Derivatives'''
  
Line 43: Line 65:
 
*'''Extrema of Multivariate Functions'''
 
*'''Extrema of Multivariate Functions'''
  
−
*'''Optimization with Constraints: Langrage Multipliers'''
+
*'''Optimization with Constraints: Lagrange Multipliers'''
  
−
===Lecture 4===
+
===[[Calc3.4|Lecture 4]]===
−
*'''The Tangent Plane'''
+
*'''Velocity and Acceleration'''
  
−
*'''The Frenet Frame'''
+
*'''Tangents and Normals'''
  
−
*'''Curvature of a Curve'''
+
*'''Binormals, Curvature, and Torison'''
  
−
===Lecture 5===
+
*'''Surface Normals'''
−
*'''The Double Integral'''
 
  
−
*'''Double Integrals in Different Coordinate Systems'''
+
===[[Calc3.5|Lecture 5]]===
 +
*'''Introduction to Integrals in Multivariable Calculus'''
  
−
*'''Double Integral Applications'''
+
*'''Double and Triple Integrals'''
  
−
===Lecture 6===
+
*'''Line Integrals'''
−
*'''The Triple Integral'''
 
  
−
*'''Triple Integrals in Different Coordinate Systems'''
+
*'''Surface Integrals'''
  
−
*'''Triple Integral Applications'''
+
*'''Integral Applications'''
−
 
 
−
===Lecture 7===
 
−
*'''Review:Vector Functions'''
 
−
 
 
−
*'''Review:Matrix Algebra'''
 
  
 +
===[[Calc3.6|Lecture 6]]===
 
*'''The Divergence'''
 
*'''The Divergence'''
  
Line 77: Line 94:
 
*'''Applications of the Divergence and Curl'''
 
*'''Applications of the Divergence and Curl'''
  
−
===Lecture 8===
+
===[[Calc3.7|Lecture 7]]===
−
*'''Generalized Integration'''
 
−
 
 
−
*'''The Line Integral'''
 
−
 
 
−
*'''Green's Theorem in the Plane'''
 
−
 
 
−
*'''The Surface Integral'''
 
−
 
 
−
===Lecture 9===
 
 
*'''Greens Theorem'''
 
*'''Greens Theorem'''
  
Line 95: Line 103:
 
*'''Applications'''
 
*'''Applications'''
  
−
===Lecture 10===
+
===[[Calc3.8|Lecture 8]]===
−
*'''The Exterior Derivative'''
+
 
  
 
*'''Applications: The Maxwell Equations'''
 
*'''Applications: The Maxwell Equations'''
  
−
*'''Generalized Stokes Theorem'''
+
===[[Calc3.9|Lecture 9]]===
−
 
+
*'''Review: Ordinary Differential Equations'''
−
===Lecture 11===
 
−
*'''Review:Ordinary Differential Equations'''
 
  
 
*'''Linear Partial Differential Equations'''
 
*'''Linear Partial Differential Equations'''
  
−
*'''The Cauchy-Riemann Operator'''
+
===[[Calc3.10|Lecture 10]]===
 +
*'''The Heat Equation'''
  
−
===Lecture 12===
+
===[[Calc3.11|Lecture 11]]===
 
*'''The Wave Equation'''
 
*'''The Wave Equation'''
  
−
===Lecture 13===
+
===[[Calc3.12|Lecture 12]]===
−
*'''The Heat Equation'''
+
 
 +
 
 +
===[[Calc3.13|Lecture 13]]===
 +
 
  
−
===Lecture 14===
+
===[[Calc3.14|Lecture 14]]===
 
*'''Course Review'''
 
*'''Course Review'''
−
 
  
 
=Single Variable Review=
 
=Single Variable Review=
 +
[[Image:Continuity.jpg|400px|thumb|A continuous function and two discontinuous functions. For more on the third example, see [[Essential Discontinuity]]]]
 
This section is not intended to be a course in elementary calculus - the student is expected to understand this material before beginning the course.  It is provided here as a reference for student.
 
This section is not intended to be a course in elementary calculus - the student is expected to understand this material before beginning the course.  It is provided here as a reference for student.
  
 
We begin with a review of functions.  Functions, in the most general sense, are rules which associate one thing with another.  For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures.  In elementary calculus, the domain and the range tend to be subsets of the real numbers.  In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets.
 
We begin with a review of functions.  Functions, in the most general sense, are rules which associate one thing with another.  For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures.  In elementary calculus, the domain and the range tend to be subsets of the real numbers.  In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets.
 +
 +
Functions which take a real numbers, and output a real number, are said to be '''continuous''' if, basically, they are connected.  At right are illustrations of continuous and discontinuous functions.  A strict definition of continuity is usually saved for a course in [[real analysis|Real Analysis]].
  
 
The '''limit''' of a function is defined so that <math>\lim_{x\rightarrow x_0} f(x) = L \ </math> if, for any <math>\epsilon>0 \ </math>, there is an <math>\delta>0 \ </math> such that every number within <math>\delta \ </math> of <math>x_0 \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>.
 
The '''limit''' of a function is defined so that <math>\lim_{x\rightarrow x_0} f(x) = L \ </math> if, for any <math>\epsilon>0 \ </math>, there is an <math>\delta>0 \ </math> such that every number within <math>\delta \ </math> of <math>x_0 \ </math> is taken by the function to a number within <math>\epsilon \ </math> of <math>L \ </math>.
Line 146: Line 158:
 
The '''indefinite integral''' or '''antiderivative''' of a function <math>f(x) \ </math> is any function <math>g(x) \ </math> such that <math>\frac{d}{dx}g(x) = f(x) \ </math>.
 
The '''indefinite integral''' or '''antiderivative''' of a function <math>f(x) \ </math> is any function <math>g(x) \ </math> such that <math>\frac{d}{dx}g(x) = f(x) \ </math>.
  
−
The '''definite integral''', or as it is defined in elementary calculus, the '''Riemann sum''', is a quantity ''EXPAND THIS''
+
The '''definite integral''' is a very important concept for students wishing to learn multivariate calculus, as we define many different kinds of definite integrals in this course. 
 +
 
 +
[[Image:Riemann.gif|right|thumb|As the division of the interval grows finer, the Riemann sum approaches a limit.]]
 +
Let <math>f(x) \ </math> be any function associating real numbers with real numbers, and let the [[Closed set|closed interval]] <math>[a,b] \ </math> be in the domain of the function.  Let's divide this interval into <math>n \ </math> smaller intervals, so that the first one is <math>[a,q_1] \ </math> and the last one is <math>[q_{n-1},b] \ </math>, and for convenience we'll call <math>a = q_0, b=q_n \ </math>.  Furthermore, let's pick a point in each of these intervals, say <math>x_j \in [q_{n-1},q_n] \ </math>.  Now we form the sum
 +
 
 +
<math>\sum_{j=1}^n f(x_j) (q_j-q_{j-1}) \ </math>
 +
 
 +
This is called a '''Riemann sum''', named after the famous 19th century mathematician [[Bernhard Riemann]].  If this sum approaches a limit as <math>n \rightarrow \ </math>[[Infinity|<math>\infty \ </math>]] no matter how we choose our <math>q_j \ </math>'s or our <math>x_j \ </math>'s, then <math>f \ </math> is said to be '''Riemann integrable''' and that limit is called the definite integral of <math>f \ </math> from <math>a \ </math> to <math>b \ </math>, and this is written <math>\int_a^b f(x)dx \ </math>
  
 
The '''fundamental theorem of calculus''' states that <math>\int_{x_o}^x \frac{d}{dx}f(x)dx = f(x) \ </math>.
 
The '''fundamental theorem of calculus''' states that <math>\int_{x_o}^x \frac{d}{dx}f(x)dx = f(x) \ </math>.
Line 153: Line 172:
  
 
=Geometry in Three Dimensions=
 
=Geometry in Three Dimensions=
−
 
+
[[Image:Xyz_right.jpg|200px|right|thumb|The right-handed xyz coordinate system.]]
 
In three dimensions, we most often work with the xyz coordinate system.  This system is called "right handed," because if we take our right palm, with our knuckles pointing in the positive x direction, and bend our fingers so that they point in the positive y direction, our thumbs automatically point in the positive z direction.  See illustration.
 
In three dimensions, we most often work with the xyz coordinate system.  This system is called "right handed," because if we take our right palm, with our knuckles pointing in the positive x direction, and bend our fingers so that they point in the positive y direction, our thumbs automatically point in the positive z direction.  See illustration.
  
 
We are familiar with the basic geometry of the plane already, but a brief review will help illustrate the concepts we encounter in the next dimension.
 
We are familiar with the basic geometry of the plane already, but a brief review will help illustrate the concepts we encounter in the next dimension.
  
−
Any line in the xy plane can be described as <math>y=ax+b \ </math>, where <math>a \ </math> is the "slope" and <math>b \ </math> is the y-intercept.  When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: <math>z=ax+by+c \ </math>, where <math>a \ </math> is the slope along the x-axis, etc.  See illustration.
+
Any line in the xy plane can be described as <math>y=ax+b \ </math>, where <math>a \ </math> is the "slope" and <math>b \ </math> is the y-intercept.  When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: <math>z=ax+by+c \ </math>, where <math>a \ </math> is the slope along the x-axis, etc.   
 +
 
 +
Describing a line in three dimensions is trickier, and requires what is called a '''parametric equation'''.  This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space.  For example, if a point is at some location <math>(x_0, y_0, z_0) \ </math> at time <math>t=0 \ </math>, and then over 1 second travels <math>a \ </math> feet in the x direction, <math>b \ </math> feet in the y direction, and <math>c \ </math> feet in the z direction, we can describe its location at any time <math>t \ </math> as <math>(x_0+at, y_0+bt, z_0+ct) \ </math>.  As it travels, this point traces out a line.
  
−
Describing a line in three dimensions is trickier, and requires what is called a '''parametric equation'''.  This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space.  For example, if a point is at some location <math>(x_0, y_0, z_0) \ </math> at time <math>t=0 \ </math>, and then over 1 second travels <math>a \ </math> feet in the x direction, <math>b \ </math> feet in the y direction, and <math>c \ </math> feet in the z direction, we can descrive its location at any time <math>t \ </math> as <math>(x_0+at, y_0+bt, z_0+ct) \ </math>.  As it travels, this point traces out a line.
+
Other geometric equations from two dimensions will find familiar analogues in three dimensions.  For example, recall the formula for a circle centered at the origin in the plane, <math>x^2 + y^2 = r^2 \ </math>, where <math>r \ </math> is the radius of the circle.  It should come as no shock to learn the formula for a sphere in three dimensions is <math>x^2 + y^2 + z^2 = r^2 \ </math>.  Other familiar geometric curves from the plane find analogues in three dimensions as well:  just as the shape described by <math>y=x^2 \ </math> in the plane is called a '''parabola''', so the surface described by <math>z=x^2 + y^2 \ </math> is called a '''parabaloid'''.  At this time, it is not necessary to discuss all of the various surfaces which appear often - the basic forms are below, and the exercises here and in future lectures will be sufficient to introduce the student to them.
 +
 
 +
[[Image:Quadrics.jpg|center|900px|thumb|Some common surfaces found in three dimensional space.  In the equations given, multiplying any coordinate by a constant yields a similar surface.]]
  
 
=Scalar Functions=
 
=Scalar Functions=
  
−
Scalar functions are the kinds of functions you're used to, but in more dimensions.  The example given earlier of a
+
Scalar functions are the kinds of functions you're used to, but in more dimensions.  The key property of a scalar function is that it returns a number.  The example given earlier of temperature at a point on the Earth is a scalar function, because even though the inputs are not numbers (they are locations, e.g., New York), the output is a temperature (e.g., 35). 
 +
 
 +
Frequently in multivariate calculus, we encounter functions which take as their inputs points in the plane or points in three dimensional space, but return numbers.  These are scalar functions.  A typical example would be a function which describes the pressure at any point in a fluid.  However, a function which takes a point in a fluid and returns the direction of the flow at that point would NOT be scalar function because the information returned it not a number - we will examine these kinds of functions in the next lecture.
 +
[[Image:Levelsurfaces.jpg|right|150px|thumb|Level surfaces of a scalar function in three dimensions.]]
 +
Scalar functions on the plane are easily visualized.  For example, a scalar function on the plane which takes a point <math>(x,y) \ </math> and returns a number <math>z \ </math> can be visualized by constructing a surface consisting of points in three dimensional space <math>(x,y,z) \ </math>.  See illustration at right.
 +
 
 +
Scalar functions which take points in space as the input are not so easily visualized, but not less useful.  For example, while we might visualize the temperature at a point on a hot plate by constructing a rubber surface above it which is raised at the hottest points, visualizing the temperature at points in our atmosphere, which differs at different altitudes, is a horse of another color, but absolutely essential to weather prediction and climatology. 
 +
 
 +
One way of visualizing scalar functions in three dimensions is to construct '''level surfaces'''.  Level surfaces are three-dimensional generalization of [[level curve]]s, which are not unlike a [[Topographic Map|topographic map]].  Level surfaces are a set of surfaces in space which are constructed so that at each point on the surface, the value of the scalar function is precisely the same as it is everywhere else on the surface.  See illustration, right.
 +
 
 +
=Problems=
 +
 
 +
==Review Problems==
 +
 
 +
'''1.''' Using the definition of "limit" given above, show that <math>\lim_{x \rightarrow x_0} ax = x_0 \ </math>, assuming <math>a \ </math> is a constant.
 +
 
 +
 
 +
'''2.''' Under the same assumptions as above, show that <math>\lim_{x \rightarrow x_0} ax^n = ax_0^n \ </math>
 +
 
 +
 
 +
'''3.''' Use [[Proof by induction|mathematical induction]] to show that <math>\lim_{x \rightarrow x_0} a_n x^n + ... + a_1 x + a_0 = a_n x_0^n + ... + a_0 \ </math>. 
 +
 
 +
You may assume that for any functions with valid limits <math>A, B, \ </math>
 +
 
 +
<math>\lim_{x \rightarrow x_0} (A(x)+B(x)) = \lim_{x \rightarrow x_0} A(x)+ \lim_{x \rightarrow x_0} B(x) \ </math>
 +
 
 +
and <math>\lim_{x \rightarrow x_0} (A(x)B(x)) = \lim_{x \rightarrow x_0} A(x) \times \lim_{x \rightarrow x_0} B(x) \ </math>.
 +
 
 +
 
 +
'''4.''' Using the definition of "limit" given above, find <math>\lim_{x \rightarrow 0} \frac{\sin(x)}{x} \ </math>.
 +
 
 +
 
 +
'''5.''' Using the limit rules given in question 3 and the definitions of derivative, prove
 +
 
 +
<math>\frac{d}{dx} (f(x)+g(x)) = f'(x) + g'(x) \ </math>
 +
 
 +
and <math>\frac{d}{dx} x^n = nx^{n-1} \ </math>
 +
 
 +
 
 +
'''6.''' Find the derivative of <math>5+e^x-2ax^3 + \sin(x^2) \ </math>, where <math>a \ </math> is a constant.
 +
 
 +
 
 +
'''7.''' Give an antiderivative of <math>\log(x) + 3x^2 \ </math>
 +
 
 +
 
 +
'''8.''' Find <math>\int_0^1 \sqrt{1-x^2}dx \ </math>.  You may consult a [[Table of Indefinite Integrals|table of integrals]].
 +
 
 +
 
 +
'''9.''' Find <math>\int_1^{e^{\pi-2}} \sin(2+\log(x)) dx \ </math>
 +
 
 +
==Main Problems==
 +
'''1.''' What sort of objects are described by these equations:
 +
 
 +
<math>z=3x+\pi y - \frac{1}{7} \ </math>
 +
 
 +
<math>y=z^2+2y^2 \ </math>
 +
 
 +
<math>x=t, y=3t-1, z=t+2 \ </math>
 +
 
 +
 
 +
'''2.''' Given what you know about plane geometry, describe the shapes described by
 +
 
 +
<math>\frac{x^2}{4} + y^2 + \frac{z^2}{9} = 1 \ </math>
 +
 
 +
<math>x^2 + {y^2 \over 16} - {z^2 \over 4} = 1 \ </math>
 +
 
 +
<math>- {x^2 \over 9} - y^2 + z^2 = 1 \ </math>
 +
 
 +
 
 +
'''3.''' Consider the shape described by <math>z=-2x+\frac{y}{2} \ </math> and <math>x=t, y=\frac{t}{3}, z=-t \ </math>.  What are these two shapes?  Do they intersect?  Where?
 +
 
 +
 
 +
 
 +
'''4.''' Sketch some level curves in the plane of <math>f(x,y) = \frac{\cos(x^2 + y^2)}{2} \ </math>.  Sketch the surface <math>z=f(x,y) \ </math>.
 +
 
 +
==Challenging Problems==
 +
Not only lines can be described with equations like <math>x=x(t), y=y(t), z=z(t) \ </math>.  When the coordinates are not linear functions of <math>t \ </math>, these equations describe a curve.  These kinds of equations are called parametric, because instead of describing shapes with only the coordinates <math>x,y,z \ </math>, they use a '''parameter,''' in this case, <math>t \ </math>. We'll discuss this more in later lectures.
 +
 
 +
'''1.''' Describe the curve <math>x=\cos(t), y=\sin(t), z=t \ </math>.  What shape does this curve have?
 +
 
 +
 
 +
'''2.''' Curves are frequently encountered when two surfaces intersect.  Do the surfaces <math>z=x^2-y^2 \ </math> and <math>x^2 + y^2 - z^2 = 1 \ </math> intersect?  Can you give a parametric description of the curve of intersection??
 +
 
 +
{{MVCalc}}

Latest revision as of 20:54, June 2, 2010

Introduction and Prerequisites

Welcome to Multivariable Calculus! This course will move quickly, and cover a lot of ground. For this reason, the instructors personal email and phone number will be available to students. We will begin with a discussion of functions of multiple variables, including differentiation and integration, and move on to more complicated matters associated with these functions, such as optimization with constraints, surface integrals, and the basics of differential geometry. It will cover the same material as a University-level Calculus 3 course, but we will begin with a review of concepts from Calculus 2 that the student should already be familiar with.

It is essential that the student have firm grasp of concepts from Calculus 1. This includes, but is not limited to,

  • Limits of functions of a single variable, including L'Hôpitals rule,
  • Differentiation of a function of a single variable,
  • Integration of a function of a single variable, including integration by parts,
  • The fundamental theorem of calculus
  • Analytic geometry of the plane, including arc length calculations
  • Infinite series, including the classic convergence tests and methods of calculation

The problems are absolutely essential to success in the course. It is by doing exercises that the student gains an understanding of a mathematical concept, not by rote memorization of a definition. Additionally, many peripheral concepts are discussed in the problems which are not touched upon, or only briefly discussed, in the main body of the lecture. Students are encouraged to do all the review exercises and all the of the main exercises, but those students who find the review material unnecessary may instead choose to complete the challenging exercises.

Course Outline

Lecture 1

  • Introduction and Course Outline
  • Review: Single variable calculus
  • Geometry in Three Dimensions
  • Scalar Functions in Two and Three Dimensions

Lecture 2

  • Vectors, and Vector Functions
  • Coordinate Systems
  • Limits of Multivariate Functions
  • Partial Derivatives
  • The Jacobian
  • Double and Triple Integrals

Lecture 3

  • Directional Derivatives
  • The Gradient
  • Parametric Curves and Surfaces
  • Extrema of Multivariate Functions
  • Optimization with Constraints: Lagrange Multipliers

Lecture 4

  • Velocity and Acceleration
  • Tangents and Normals
  • Binormals, Curvature, and Torison
  • Surface Normals

Lecture 5

  • Introduction to Integrals in Multivariable Calculus
  • Double and Triple Integrals
  • Line Integrals
  • Surface Integrals
  • Integral Applications

Lecture 6

  • The Divergence
  • The Curl
  • Applications of the Divergence and Curl

Lecture 7

  • Greens Theorem
  • Gauss' Theorem
  • Stokes Theorem
  • Applications

Lecture 8

  • Applications: The Maxwell Equations

Lecture 9

  • Review: Ordinary Differential Equations
  • Linear Partial Differential Equations

Lecture 10

  • The Heat Equation

Lecture 11

  • The Wave Equation

Lecture 12

Lecture 13

Lecture 14

  • Course Review

Single Variable Review

A continuous function and two discontinuous functions. For more on the third example, see Essential Discontinuity

This section is not intended to be a course in elementary calculus - the student is expected to understand this material before beginning the course. It is provided here as a reference for student.

We begin with a review of functions. Functions, in the most general sense, are rules which associate one thing with another. For example, there is a function which takes a place on Earth and returns its temperature - in that case, the "domain" of the function is the Earth, and the "range" are temperatures. In elementary calculus, the domain and the range tend to be subsets of the real numbers. In this course, the domain and range may be real numbers, or sets of vectors in higher dimensions, or matrix sets.

Functions which take a real numbers, and output a real number, are said to be continuous if, basically, they are connected. At right are illustrations of continuous and discontinuous functions. A strict definition of continuity is usually saved for a course in Real Analysis.

The limit of a function is defined so that if, for any , there is an such that every number within of is taken by the function to a number within of .

The derivative of a function is the limit and is the slope of the line tangent to the function's curve at . To write the derivative of a function with respect to , we write or .

Some special derivative bear reviewing:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} f(x)g(x) = f'(x)g(x) + f(x)g'(x) }

The indefinite integral or antiderivative of a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \ } is any function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) \ } such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}g(x) = f(x) \ } .

The definite integral is a very important concept for students wishing to learn multivariate calculus, as we define many different kinds of definite integrals in this course.

As the division of the interval grows finer, the Riemann sum approaches a limit.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \ } be any function associating real numbers with real numbers, and let the closed interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b] \ } be in the domain of the function. Let's divide this interval into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \ } smaller intervals, so that the first one is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,q_1] \ } and the last one is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [q_{n-1},b] \ } , and for convenience we'll call Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = q_0, b=q_n \ } . Furthermore, let's pick a point in each of these intervals, say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j \in [q_{n-1},q_n] \ } . Now we form the sum

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j=1}^n f(x_j) (q_j-q_{j-1}) \ }

This is called a Riemann sum, named after the famous 19th century mathematician Bernhard Riemann. If this sum approaches a limit as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \rightarrow \ } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty \ } no matter how we choose our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_j \ } 's or our Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j \ } 's, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \ } is said to be Riemann integrable and that limit is called the definite integral of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \ } from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \ } , and this is written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b f(x)dx \ }

The fundamental theorem of calculus states that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{x_o}^x \frac{d}{dx}f(x)dx = f(x) \ } .

The material just described should be so familiar to the student that it can be recited as easily as multiplication tables or trigonometric values on the unit circle.

Geometry in Three Dimensions

The right-handed xyz coordinate system.

In three dimensions, we most often work with the xyz coordinate system. This system is called "right handed," because if we take our right palm, with our knuckles pointing in the positive x direction, and bend our fingers so that they point in the positive y direction, our thumbs automatically point in the positive z direction. See illustration.

We are familiar with the basic geometry of the plane already, but a brief review will help illustrate the concepts we encounter in the next dimension.

Any line in the xy plane can be described as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=ax+b \ } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ } is the "slope" and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \ } is the y-intercept. When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=ax+by+c \ } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ } is the slope along the x-axis, etc.

Describing a line in three dimensions is trickier, and requires what is called a parametric equation. This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space. For example, if a point is at some location Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0, y_0, z_0) \ } at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0 \ } , and then over 1 second travels Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ } feet in the x direction, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \ } feet in the y direction, and feet in the z direction, we can describe its location at any time as . As it travels, this point traces out a line.

Other geometric equations from two dimensions will find familiar analogues in three dimensions. For example, recall the formula for a circle centered at the origin in the plane, , where is the radius of the circle. It should come as no shock to learn the formula for a sphere in three dimensions is . Other familiar geometric curves from the plane find analogues in three dimensions as well: just as the shape described by in the plane is called a parabola, so the surface described by is called a parabaloid. At this time, it is not necessary to discuss all of the various surfaces which appear often - the basic forms are below, and the exercises here and in future lectures will be sufficient to introduce the student to them.

Some common surfaces found in three dimensional space. In the equations given, multiplying any coordinate by a constant yields a similar surface.

Scalar Functions

Scalar functions are the kinds of functions you're used to, but in more dimensions. The key property of a scalar function is that it returns a number. The example given earlier of temperature at a point on the Earth is a scalar function, because even though the inputs are not numbers (they are locations, e.g., New York), the output is a temperature (e.g., 35).

Frequently in multivariate calculus, we encounter functions which take as their inputs points in the plane or points in three dimensional space, but return numbers. These are scalar functions. A typical example would be a function which describes the pressure at any point in a fluid. However, a function which takes a point in a fluid and returns the direction of the flow at that point would NOT be scalar function because the information returned it not a number - we will examine these kinds of functions in the next lecture.

Level surfaces of a scalar function in three dimensions.

Scalar functions on the plane are easily visualized. For example, a scalar function on the plane which takes a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y) \ } and returns a number can be visualized by constructing a surface consisting of points in three dimensional space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z) \ } . See illustration at right.

Scalar functions which take points in space as the input are not so easily visualized, but not less useful. For example, while we might visualize the temperature at a point on a hot plate by constructing a rubber surface above it which is raised at the hottest points, visualizing the temperature at points in our atmosphere, which differs at different altitudes, is a horse of another color, but absolutely essential to weather prediction and climatology.

One way of visualizing scalar functions in three dimensions is to construct level surfaces. Level surfaces are three-dimensional generalization of level curves, which are not unlike a topographic map. Level surfaces are a set of surfaces in space which are constructed so that at each point on the surface, the value of the scalar function is precisely the same as it is everywhere else on the surface. See illustration, right.

Problems

Review Problems

1. Using the definition of "limit" given above, show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow x_0} ax = x_0 \ } , assuming Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ } is a constant.


2. Under the same assumptions as above, show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow x_0} ax^n = ax_0^n \ }


3. Use mathematical induction to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow x_0} a_n x^n + ... + a_1 x + a_0 = a_n x_0^n + ... + a_0 \ } .

You may assume that for any functions with valid limits Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A, B, \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow x_0} (A(x)+B(x)) = \lim_{x \rightarrow x_0} A(x)+ \lim_{x \rightarrow x_0} B(x) \ }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow x_0} (A(x)B(x)) = \lim_{x \rightarrow x_0} A(x) \times \lim_{x \rightarrow x_0} B(x) \ } .


4. Using the definition of "limit" given above, find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \rightarrow 0} \frac{\sin(x)}{x} \ } .


5. Using the limit rules given in question 3 and the definitions of derivative, prove

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} (f(x)+g(x)) = f'(x) + g'(x) \ }

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx} x^n = nx^{n-1} \ }


6. Find the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5+e^x-2ax^3 + \sin(x^2) \ } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ } is a constant.


7. Give an antiderivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log(x) + 3x^2 \ }


8. Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^1 \sqrt{1-x^2}dx \ } . You may consult a table of integrals.


9. Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_1^{e^{\pi-2}} \sin(2+\log(x)) dx \ }

Main Problems

1. What sort of objects are described by these equations:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=3x+\pi y - \frac{1}{7} \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=z^2+2y^2 \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=t, y=3t-1, z=t+2 \ }


2. Given what you know about plane geometry, describe the shapes described by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x^2}{4} + y^2 + \frac{z^2}{9} = 1 \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + {y^2 \over 16} - {z^2 \over 4} = 1 \ }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - {x^2 \over 9} - y^2 + z^2 = 1 \ }


3. Consider the shape described by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=-2x+\frac{y}{2} \ } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=t, y=\frac{t}{3}, z=-t \ } . What are these two shapes? Do they intersect? Where?


4. Sketch some level curves in the plane of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y) = \frac{\cos(x^2 + y^2)}{2} \ } . Sketch the surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=f(x,y) \ } .

Challenging Problems

Not only lines can be described with equations like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x(t), y=y(t), z=z(t) \ } . When the coordinates are not linear functions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \ } , these equations describe a curve. These kinds of equations are called parametric, because instead of describing shapes with only the coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y,z \ } , they use a parameter, in this case, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \ } . We'll discuss this more in later lectures.

1. Describe the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\cos(t), y=\sin(t), z=t \ } . What shape does this curve have?


2. Curves are frequently encountered when two surfaces intersect. Do the surfaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=x^2-y^2 \ } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + y^2 - z^2 = 1 \ } intersect? Can you give a parametric description of the curve of intersection??

v • d • e
Multivariable Calculus Course
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