1. 講者： 許永昌 老師 1

2. Contents Find the direction of the maximum change of temperature. Partial Derivative Gradient as a Vector Operator Example p41e (P34) A Geometrical Interpretation Summary 2

3. Find the direction of the maximum change of temperature df = f(x+dx, y+dy, z+dz) – f(x, y, z) Restriction: Finding: max|df| Example: T=x 2 +xy+y 2. Code: grad_of_T.mgrad_of_T.m 3 要證 要講

4. Partial Derivative For 2D: E.g. Partial derivative is very important in many field, such as mechanics, electrodynamics, engineering, thermodynamics, and etc. E.g. dU=  PdV+TdS, we get. 4 It means that S is fixed for this partial derivative. y

5. Total Variation of a function For 2D: df(x, y) = Proof: For higher dimension: 5 A B C fBfAfBfA fCfAfCfA fBfCfBfC

6. Assume f=x(x+y)=xz, i.e. z  x+y, what is the function of ? Confuse? Read P4 of this pptx file. What is df? 6

7. Gradient as a Vector Operator Total variation: For 3D: df(x, y, z)= Displacement dr=(x+dx, y+dy, z+dz) – (x, y, z) = (dx, dy, dz). 7

8. Gradient Is a vector operator:  = Is the normal to the equipotential surface A surface defined by  (x, y, z)=C, we get d  =0. Besides, d  =  dr, it means that dr should obeys  dr =0, i.e.  | P  is the normal of  (x, y, z)=C surface at point p. In physics and engineering, a force which can be written as F=  V is a conservative force.  F conservative  dr=   dV=V i  V f. It is independent of the path. 8

9. Example P41e (P34) The Gradient of a Function of r. 9

10. Summary 10

11. Homework 1.5.2e (1.6.3) 1.5.4e (1.6.5) 11

12. Nouns 12