2. (10 pts) Short answers: answer the following questions in no more than two li

Question

2. (10 pts) Short answers: answer the following questions in no more than two lines of text (a) If you know that lim ,ma)li ),what can you conclude about the conti- (b) Let f be a scalar field and F a vector field. Which of the following expressions are and/or calculations. nuity of f(, y) at (0,0)? meaningful? (i)VF (iv)V(div F) (ii) div f (vi) curl(curl F) (ii) curl F (v) curlVf) (c) Write down the formula for the average value of a scalar function f(x, y, z) over a solid region E in space (d) If a transformation T is given by -u -,y u, what is the Jacobian of T? (e) If all component functions of F have continuous partial derivatives on R3, and if F d 0 for any closed space curve C, what can you say about F? w

Explanation / Answer

We can conclude that it is continuous at (0,0)

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b)
f = scalar
F = vector

We can find the grad for a scalar field
we can find div for vector field
and we can find curl for a vector field

grad F ---> no meaning
curl F --- meaningful
div f ---> no meaning
grd (div F) --> meaningful
curl (grad f) ---> no meaning
curl (curl F) --> meaningful

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c)
This is just the 1/(volume of E) * (triple integral f(x,y,z) dV

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d)
x = u - v
y = uv

Finding partials,
xu = 1 , xv = -1
yu = v , yv = u

Now, jacobian has formula
|xu*yv - xv*yu|

|u + v|

u + v ---> ANS

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e)
It'd mean that the function F is path independent
as in we can find a potential function, f for vector field F.

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