Need help with my Calculus question – I’m studying for my class.

The questions below are about Vector Calculus and Fourier Series.

*1. Consider the vector field:

F(x, y, z) := 1 i + 2y j +

1. By direct computation show that ? × F = 0.
2. Find a potential function for F.

2z k. 1+x+z2 +y2

1+x+z2 +y2 1+x+z2 +y2

c. How does the result in Part a. follow from Part b.?
d. Find??CF·dr,whereCisacurvefrom0=(0,0,0)tothepoint

(1, 1, 2).

*4. Starting with the divergence theorem and two smooth scalar functions f and g, a closed surface S enclosing a volume V , derive the first and the second Green’s formulae:

a. Green’s first formula:
???? (f?g)·ndS=?????? ??f?2g+?f·?g??dV.

SV

b. Green’s second formula:

???? (f?g)·ndS????? (g?f)·ndS=?????? ??f?2g?g?2f??dV.

*6. Let m, n be positive integers. Verify the integral identities:

and

1 ?? L ??n?x?? ??m?x??
L cos L cos L dx=?mn

?L

1 ?? L ??n?x?? ??m?x??
L cos L sin L dx=0.