As in single variable calculus, there is a multivariable chain rule. The version
with several variables is more complicated and we will use the tangent
approximation and total differentials to help understand and organize it.
Also related to the tangent approximation formula is the gradient of a function.
The gradient is one of the key concepts in multivariable calculus. It is a
vector field, so it allows us to use vector techniques to study functions of
several variables. Geometrically, it is perpendicular to the level curves or
surfaces and represents the direction of most rapid change of the function.
Analytically, it holds all the rate information for the function and can be used
to compute the rate of change in any direction.
Last Modification: 2020-09-17 Thu 12:19
- Captured On
- [2020-01-18 Sat]
- Source
- Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare
Captured On [2020-02-06 Thu 12:28] Source Session 32: Total Differentials and the Chain Rule | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Differentials and implicit differentiation
Figure 2: Example of implicit differentiation
Figure 3: Total differential
Figure 4: What can do with differentials
Figure 5: What can do with differentials (2)
Figure 6: Chain Rule...
Captured On [2020-02-06 Thu 12:28] Source Session 34: The Chain Rule with More Variables | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Chain rule with more variables
Figure 2: Expansion of chain rule
Figure 3: Partial derivative with more variables
Figure 4: Example with polar coordinates
2 How is the chain rule for partial derivative with more than one independent variable?...
Captured On [2020-02-06 Thu 12:28] Source Session 35: Gradient: Definition, Perpendicular to Level Curves | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall chain rule
Figure 2: Definition of gradient
Figure 3: Gradient is perpendicular to level surface
Figure 4: Example of Gradient (1)
Figure 5: Example of Gradient (2)
2 What is the definition of gradient?...
Captured On [2020-02-06 Thu 12:29] Source Session 36: Proof | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Curve that stays on the level \(w=c\)
Figure 2: Chain rule
Figure 3: Gradient perpendicular to tangent vector
Figure 4: Any vector of tangent plane is perpendicular to gradient
2 How can we proof that gradient is perpendicular to level curves and surfaces?...
Captured On [2020-02-06 Thu 12:29] Source Session 37: Example | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Find tangent plane to surface
Figure 2: Gradient is the normal vector of tangent plane
Figure 3: Another way throught linear approximation
Figure 4: Meaning of approximation
2 How can we proof that gradient is perpendicular to level curves and surfaces?...
Captured On [2020-02-06 Thu 12:29] Source Session 38: Directional Derivatives | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Directional derivatives
Figure 2: Straight line trayectory
Figure 3: Directional derivative in direction of \(\vu{u}\)
Figure 4: Slope of a slice
Figure 5: Directional derivative as a components of gradient
Figure 6: Geometrically
Figure 7: Fastest increase of \(w\)...