Dr.-Ing. Erwin Sitompul President University Lecture 6 Multivariable Calculus President UniversityErwin SitompulMVC 6/1

President UniversityErwin SitompulMVC 6/2 Directional Derivatives in the Plane Chapter 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.

President UniversityErwin SitompulMVC 6/3 Directional Derivatives in the Plane Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/4 Directional Derivatives in the Plane Example Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/5 Interpretation of the Directional Derivative Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/6 Calculation and Gradients Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/7 Calculation and Gradients Chapter Directional Derivatives and Gradient Vectors Example

President UniversityErwin SitompulMVC 6/8 Calculation and Gradients Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/9 Calculation and Gradients Chapter Directional Derivatives and Gradient Vectors Example

President UniversityErwin SitompulMVC 6/10 Gradients and Tangents to Level Curves Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/11 Gradients and Tangents to Level Curves Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/12 Gradients and Tangents to Level Curves Chapter Directional Derivatives and Gradient Vectors Example Any other way to find the equation of the tangent line?

President UniversityErwin SitompulMVC 6/13 Gradients and Tangents to Level Curves Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/14 Gradients and Tangents to Level Curves Chapter Directional Derivatives and Gradient Vectors Example

President UniversityErwin SitompulMVC 6/15 Functions of Three Variables Chapter Directional Derivatives and Gradient Vectors

President UniversityErwin SitompulMVC 6/16 Functions of Three Variables Chapter Directional Derivatives and Gradient Vectors Example

President UniversityErwin SitompulMVC 6/17 Functions of Three Variables Chapter Directional Derivatives and Gradient Vectors Example

President UniversityErwin SitompulMVC 6/18 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 Chapter Directional Derivatives and Gradient Vectors