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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.

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directional derivatives in the plane

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.
directional derivatives in the plane1

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Directional Derivatives in the Plane
directional derivatives in the plane2

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Directional Derivatives in the Plane
  • Example
interpretation of the directional derivative

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Interpretation of the Directional Derivative
calculation and gradients

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Calculation and Gradients
calculation and gradients1

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Calculation and Gradients
  • Example
calculation and gradients2

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Calculation and Gradients
calculation and gradients3

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Calculation and Gradients
  • Example
gradients and tangents to level curves

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Gradients and Tangents to Level Curves
gradients and tangents to level curves1

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Gradients and Tangents to Level Curves
gradients and tangents to level curves2

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?
gradients and tangents to level curves3

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Gradients and Tangents to Level Curves
gradients and tangents to level curves4

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Gradients and Tangents to Level Curves
  • Example
functions of three variables

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Functions of Three Variables
functions of three variables1

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Functions of Three Variables
  • Example
functions of three variables2

Chapter 14

14.5 Directional Derivatives and Gradient Vectors

Functions of Three Variables
  • Example
homework 6

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.
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