Multivariable Calculus Lecture 6 Dr Ing Erwin Sitompul

Multivariable Calculus Lecture 6 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com President University Erwin Sitompul MVC 6/1

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Directional Derivatives in the Plane n The next map shows the contours on the area in New York. The tributary streams flow perpendicular to the contour. n The streams are following paths of steepest descent so the water reach the river as quickly as possible. n The instantaneous rate of change in a stream’s altitude above sea level has a particular direction. n In this section why this “downhill” direction is perpendicular to the contours. President University Erwin Sitompul MVC 6/2

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Directional Derivatives in the Plane President University Erwin Sitompul MVC 6/3

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Directional Derivatives in the Plane n Example President University Erwin Sitompul MVC 6/4

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Interpretation of the Directional Derivative President University Erwin Sitompul MVC 6/5

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Calculation and Gradients President University Erwin Sitompul MVC 6/6

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Calculation and Gradients n Example President University Erwin Sitompul MVC 6/7

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Calculation and Gradients President University Erwin Sitompul MVC 6/8

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Calculation and Gradients n Example President University Erwin Sitompul MVC 6/9

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Gradients and Tangents to Level Curves President University Erwin Sitompul MVC 6/10

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Gradients and Tangents to Level Curves President University Erwin Sitompul MVC 6/11

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Gradients and Tangents to Level Curves n Example n Any other way to find the equation of the tangent line? President University Erwin Sitompul MVC 6/12

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Gradients and Tangents to Level Curves President University Erwin Sitompul MVC 6/13

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Gradients and Tangents to Level Curves n Example President University Erwin Sitompul MVC 6/14

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Functions of Three Variables President University Erwin Sitompul MVC 6/15

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Functions of Three Variables n Example President University Erwin Sitompul MVC 6/16

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Functions of Three Variables n Example President University Erwin Sitompul MVC 6/17

Chapter 14 14. 5 Directional Derivatives and Gradient Vectors Homework 6 n Exercise 14. 5, No. 15. n Exercise 14. 5, No. 22. n Exercise 14. 5, No. 31. n Homeworks must not be submitted, but there will be a 15 -minute short quiz in every lecture, asking 1 -2 of the homework problems. n Next short quiz: Next week, at 17. 15. President University Erwin Sitompul MVC 6/18