Overview
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 +
- By direct computation show that ? × F = 0.
- 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.