Michael Corral: Vector Calculus

74 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES
A
For Exercises 1-16, find ∂f
∂x
and
∂f
∂y .
☛ ✟
✡Exercises
✠
1. f(x,y)= x 2 +y 2 2. f(x,y)=cos(x+y)
3. f(x,y)= √ x 2 +y+4 4. f(x,y)= x+1
y+1
5. f(x,y)=e xy + xy 6. f(x,y)= x 2 −y 2 +6xy+4x−8y+2
7. f(x,y)= x 4 8. f(x,y)= x+2y
9. f(x,y)= √ x 2 +y 2 10. f(x,y)=sin(x+y)
11. f(x,y)= 3√ x 2 +y+4 12. f(x,y)= xy+1
x+y
13. f(x,y)=e −(x2 +y 2 )
14. f(x,y)=ln(xy)
15. f(x,y)=sin(xy) 16. f(x,y)=tan(x+y)
For Exercises 17-26, find ∂2 f
, ∂2 f
and ∂2 f
∂x 2 ∂y 2 ∂y∂x
(use Exercises 1-8, 14, 15).
17. f(x,y)= x 2 +y 2 18. f(x,y)=cos(x+y)
19. f(x,y)= √ x 2 +y+4 20. f(x,y)= x+1
y+1
21. f(x,y)=e xy + xy 22. f(x,y)= x 2 −y 2 +6xy+4x−8y+2
23. f(x,y)= x 4 24. f(x,y)= x+2y
25. f(x,y)=ln(xy) 26. f(x,y)=sin(xy)
B
27. Show that the function f(x,y)=sin(x+y)+cos(x−y) satisfies the wave equation
∂ 2 f f
∂x 2−∂2 ∂y2= 0.
The wave equation is an example of a partial differential equation.
28. Let u and v be twice-differentiable functions of a single variable, and let c0
be a constant. Show that f(x,y)=u(x+cy)+v(x−cy) is a solution of the general
one-dimensional wave equation 3 ∂ 2 f 1 ∂ 2 f
∂x 2− c 2 ∂y2= 0.
3 Conversely, it turns out that any solution must be of this form. See Ch. 1 in WEINBERGER.