Solve the problem.Assuming all the necessary derivatives exist, show that if 
= 0 for all closed curves C to which Green's Theorem applies, then f satisfies the Laplace equation 
+ 
= 0 for all regions bounded by closed curves C to which Green's Theorem applies.
What will be an ideal response?
Let C be a closed curve for which Green's Theorem applies. Then 
If the Laplace equation does not hold for f on R, then there is a simple closed curve C' in the interior of R which bounds a simply connected region R' on which 
+ 
? 0 for all (x, y) in R'. Moreover, without loss of generality, we can assume that 
+ 
> 0 . Hence 
As Green's Theorem applies to C', we also have that
0 =
= 
which is a contradiction.
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Solve the problem.The range R of a projectile is related to the initial velocity v and projection angle ? by the equation 
where g is a constant. How is 
related to 
if ? is constant?
A. 
= 
B. 
= 
C. 
= 
D. 
= 
Solve the problem.Your bank account has $85 in it when you write checks for $21, $20, and $35. You then deposit $41 and $22. How much is in the account? Are you overdrawn?
A. $98, no B. $107, no C. -$98, yes D. $72, no
Solve the problem.A bag contains 7 apples and 5 oranges. If you select 6 pieces of fruit without looking, how many ways can you get 6 apples?
A. 14 B. 35 C. 12 D. 7
Convert the capacity as indicated.1
pints to cups
A. 3 c B. 4 c C. 2 c D. 6 c