18.02SC | Fall 2010 | Undergraduate

# Multivariable Calculus

2. Partial Derivatives

## Part B: Chain Rule, Gradient and Directional Derivatives

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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.

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### Overview

In this session you will:

• Do the problems
• Use the solutions to check your work

### Problem Set

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Review an example
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Total Differentials and Chain Rule

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Examples

Chain Rule and Total Differentials (PDF)

### Problems and Solutions

Problems: Chain Rule and Total Differentials (PDF)

Solutions (PDF)

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« Previous | Next »

### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Examples

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Problems and Solutions

Problems: Chain Rule (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples
• Review an example
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Chain Rule with More Variables

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Chain Rule (PDF)

### Examples

Chain Rule with more Variables (PDF)

### Problems and Solutions

Problems: Chain Rule Practice (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples

### Lecture Video

#### Video Excerpts

Clip: Definition, Perpendicular to Level Curves

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Problems and Solutions

Problems: Elliptic Paraboloid (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples
• Do problems and use solutions to check your work

### Lecture Video

#### Video Excerpts

Clip: Proof

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Proof the Gradient is Perpendicular to Level Curves and Surfaces (PDF)

### Problems and Solutions

Problems: Gradient and Level Curves (PDF)

Solutions (PDF)

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« Previous | Next »

### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Review an example
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Example

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

### Examples

Equation of the Tangent Plane Using the Gradient (PDF)

### Problems and Solutions

Problems: Equation of a Tangent Plane (PDF)

Solutions (PDF)

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### Overview

In this session you will:

• Watch a lecture video clip and read board notes
• Read course notes and examples
• Watch a recitation video

### Lecture Video

#### Video Excerpts

Clip: Directional Derivatives

The following images show the chalkboard contents from these video excerpts. Click each image to enlarge.

Directional Derivatives (PDF)

### Problems and Solutions

Problems: Directional Derivatives (PDF)

Solutions (PDF)

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## Course Info

Fall 2010
##### Learning Resource Types
Simulations
Exams with Solutions
Problem Sets with Solutions
Lecture Videos
Lecture Notes