# Directional Derivatives in the Plane - PowerPoint PPT Presentation

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Chapter 14. 14.5 Directional Derivatives and Gradient Vectors. Directional Derivatives in the Plane. The next map shows the contours on the area in New York. The tributary streams flow perpendicular to the contour.

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Directional Derivatives in the Plane

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Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Directional Derivatives in the Plane

• The next map shows the contours on the area in New York. The tributary streams flow perpendicular to the contour.

• The streams are following paths of steepest descent so the water reach the river as quickly as possible.

• The instantaneous rate of change in a stream’s altitude above sea level has a particular direction.

• In this section why this “downhill” direction is perpendicular to the contours.

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Directional Derivatives in the Plane

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Directional Derivatives in the Plane

• Example

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Interpretation of the Directional Derivative

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Calculation and Gradients

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Calculation and Gradients

• Example

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Calculation and Gradients

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Calculation and Gradients

• Example

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Gradients and Tangents to Level Curves

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Gradients and Tangents to Level Curves

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Gradients and Tangents to Level Curves

• Example

• Any other way to find the equation of the tangent line?

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Gradients and Tangents to Level Curves

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Gradients and Tangents to Level Curves

• Example

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Functions of Three Variables

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Functions of Three Variables

• Example

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Functions of Three Variables

• Example

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

### Homework 6

• Exercise 14.5, No. 15.

• Exercise 14.5, No. 22.

• Exercise 14.5, No. 31.

• Homeworks must not be submitted, but there will be a 15-minute short quiz in every lecture, asking 1-2 of the homework problems.

• Next short quiz: Next week, at 17.15.