1. Chapter 4: Partial Derivatives I. Intro Say I have a surface whose height depends on x & y I might want to know how f(x,y) changes in the y-direction if I stay at a constant x value. Partial derivative:  keep x constant and differentiate with respect to y  keep y constant and differentiate with respect to x f(x,y) x y

2. ex: Notation: Note: the order of the differentiation doesn’t matter as long as both are continuous.

3. ex: Wave:  change in space at a fixed time  change in time at a fixed location ex: Wave equation: Say describes an electromagnetic wave. Then, in order to satisfy Maxwell’s Equations, the wave must obey the wave equation: where c is the speed of light in the medium.

4. ex: Let We know that the speed of this wave is. (Physics 216!) Does this wave satisfy the wave equation?

5. II. Total Derivatives Say we have a function: y = f(t) (ex: ) We want to know how big of a change Δy we get in a small time interval Δt. Note: Δt and Δy are the exact values of the change in t and change in y. dt and dy are the approximate values of the change in t and change in y. In this case, dt = Δt but dy ≠ Δy. Let  ( is slope of tangent) Then  (dy is a good estimate for ∆y, provided dt is small.) ΔtΔt dy } ΔyΔy t y y(t) yoyo tangent line

6. Say we have a function of multiple variables: z = f(x,y,t) We want to know ∆z in response to a small change dx: If we define the total differential dz as: Then dz is a good approximation of the total change ∆z resulting from small changes to dx, dy, and dz.

7. ex: Let (two resistors in parallel) R 1 = 10Ω and R 2 = 10Ω Say R 1 is changed to 10.1Ω. By how much does R eq change?

8. ex: Same problem as above Say R 1 is changed to 10.1 Ω By how much does R 2 have to change in order to keep R eq constant?

9. III. Implicit Differentiation ex: Say I know that the trajectory of an object is given by: What is the velocity of the object at x = 1?

10. IV. Chain Rule Let f(x,y,z). Then ex: find and

11. ex: Two parallel, uniform lines of charge of ±λ C/m located at (x = ±1, y = 0) We want to solve for the E-field lines. We know: We want E-field lines, e.g. the expressions for y=f(x) that are tangent to the E-field at each point. ←slope of the tangent line -λ-λ+λ+λ x y

12. We can integrate this to find: where c is the constant of integration Note: this is the equation for a circle of radius centered at (0,c) We can vary c between -∞ to ∞ to pick off a whole family of field lines.

13. V. Differentiating Integrals A.Let Then: 1) 2) ex:

14. So, let Then ex: (preclass q)

15. B. If both integrals exist, then in general: ex: Method 1: Method 2:

16. C. Leibniz’s Rule:

17. ex: Method 1: method 2: