Automoderated users, edit
3,552
edits
Changes
Jump to navigation
Jump to search
Line 54:
Line 54: + + + + + + + + + + + + Line 61:
Line 73: − +
→Lecture 3: a start
===[[Calc3.3|Lecture 3]]===
===[[Calc3.3|Lecture 3]]===
*'''The Gradient'''
*'''The Gradient'''
We've discussed partial derivatives, but it seems there should be a way to find a derivative which is in some sense "complete." We'll address this concept for vector fields in the next lecture, but for now let's examine the gradient, a differential operator for scalar functions. For every point in a scalar field, the '''gradient''' is a vector in the direction of greatest change, with length equal to the rate of change in that direction (ie, the directional derivative).
The gradient is easily computed:
<math>\vec{grad} f = \vec{i}\partial_x f+ \vec{j}\partial_y f + \vec{k}\partial_z f</math>.
If we wanted to construct a more concise version of this, we can always form the "vector" often called '''del''' or '''nabla''':
<math>\nabla = \vec{i}\frac{\partial}{\partial x} + \vec{j}\frac{\partial}{\partial y} + \vec{k}\frac{\partial}{\partial z}</math>.
Then, if we just take this operator of ''f'', we get the gradient: <math>\vec{grad} f = \nabla f</math>. Remember this operator; we'll encounter it again.
*'''Directional Derivatives'''
*'''Directional Derivatives'''
*'''Extrema of Multivariate Functions'''
*'''Extrema of Multivariate Functions'''
*'''Optimization with Constraints: Langrage Multipliers'''
*'''Optimization with Constraints: Lagrange Multipliers'''
===[[Calc3.4|Lecture 4]]===
===[[Calc3.4|Lecture 4]]===