Changes

Nevertheless, the kinds of phenomenon we see in the multivariate case is very different.  Note that while it is still possible to define a limit from a direction, we don't bother to do so - unlike the one-variable case when a one-sided deriative told us half of the functions behavior at a point, in higher dimensions such a limit tells us almost nothing.  Indeed, the only kind of phenomenon that these kinds of limits are capable of describing are "point discontinuties," which can be "fixed" just be defining the function to be the limit at that point.

Nevertheless, the kinds of phenomenon we see in the multivariate case is very different.  Note that while it is still possible to define a limit from a direction, we don't bother to do so - unlike the one-variable case when a one-sided deriative told us half of the functions behavior at a point, in higher dimensions such a limit tells us almost nothing.  Indeed, the only kind of phenomenon that these kinds of limits are capable of describing are "point discontinuties," which can be "fixed" just be defining the function to be the limit at that point.

=[[Derivative#Partial_derivatives|Partial Derivatives]]=

=[[Derivative#Partial derivatives|Partial Derivatives]]=

:''See [[Calc3.2/Partial derivatives proof|the proof page]] for details.''

:''See [[Calc3.2/Partial derivatives proof|the proof page]] for details.''

When dealing with functions of more than one variable, one has to be careful about what a "derivative" is.  The operative concept is the ''partial derivative'', which is the derivative of the function with respect to a given argument ''while holding all the other arguments constant''.  It is written like a derivative, but with a different "d" character.

When dealing with functions of more than one variable, one has to be careful about what a "derivative" is.  The operative concept is the ''partial derivative'', which is the derivative of the function with respect to a given argument ''while holding all the other arguments constant''.  It is written like a derivative, but with a different "d" character.

Its Cartesian components are:

Its Cartesian components are:

::<math>(A_x=- r \sin \theta, A_y=r \cos \theta)\,</math>

::<math>(A_x=- r \sin \theta, A_y=r \cos \theta)\,</math>

It always points "laterally", at right angles to the radial direction.  More surprisingly, its length is <math>\sqrt{A_x^2 + A_y^2} = r\,</math>.  The angular basis vector is ''not'' a unit vector!  It is bigger when farther away from the origin, even though the sum of the squares of its components is 1.<ref>Some textbooks use a definition of vector components such that basis vectors always have length 1.  Such a treatment imposes the requirement that the coordinate lines be perpendicular everywhere.  The formulas for vector operations are more complicated and less elegant when this is done.</ref>  In general curvilinear coordinates, the "square root of the sum of the squares of the components" rule is not correct for the length of a vector.  The correct rule will be given [[Calc3.2#The_Length_of_a_Vector_in_Arbitrary_Coordinates|below]].

It always points "laterally", at right angles to the radial direction.  More surprisingly, its length is <math>\sqrt{A_x^2 + A_y^2} = r\,</math>.  The angular basis vector is ''not'' a unit vector!  It is bigger when farther away from the origin, even though the sum of the squares of its components is 1.<ref>Some textbooks use a definition of vector components such that basis vectors always have length 1.  Such a treatment imposes the requirement that the coordinate lines be perpendicular everywhere.  The formulas for vector operations are more complicated and less elegant when this is done.</ref>  In general curvilinear coordinates, the "square root of the sum of the squares of the components" rule is not correct for the length of a vector.  The correct rule will be given [[Calc3.2#The Length of a Vector in Arbitrary Coordinates|below]].

===The Dot Product in Arbitrary Coordinates===

===The Dot Product in Arbitrary Coordinates===