directional derivatives in the plane

Dr.-Ing. Erwin SitompulPresident University

Lecture 6

Multivariable Calculus

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Directional Derivatives in the PlaneChapter 14 14.5 Directional Derivatives and Gradient Vectors

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.

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Directional Derivatives in the PlaneChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Directional Derivatives in the Plane Example

Chapter 14 14.5 Directional Derivatives and Gradient Vectors

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Interpretation of the Directional DerivativeChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Calculation and GradientsChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Calculation and GradientsChapter 14 14.5 Directional Derivatives and Gradient Vectors

Example

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Calculation and GradientsChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Calculation and GradientsChapter 14 14.5 Directional Derivatives and Gradient Vectors

Example

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Gradients and Tangents to Level CurvesChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Gradients and Tangents to Level CurvesChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Gradients and Tangents to Level CurvesChapter 14 14.5 Directional Derivatives and Gradient Vectors

Example

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

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Gradients and Tangents to Level CurvesChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Gradients and Tangents to Level CurvesChapter 14 14.5 Directional Derivatives and Gradient Vectors

Example

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Functions of Three VariablesChapter 14 14.5 Directional Derivatives and Gradient Vectors

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Functions of Three VariablesChapter 14 14.5 Directional Derivatives and Gradient Vectors

Example

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Functions of Three VariablesChapter 14 14.5 Directional Derivatives and Gradient Vectors

Example

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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.

Chapter 14 14.5 Directional Derivatives and Gradient Vectors