Question

Let **F** (*x, y*) =
2*xy***i** + (*x* –
2*y*)**j**, **r** (*t*) =
sin *t***i** – 2 cos *t*
**j**, 0 ≤ *t* ≤ *π*. Then *C*
**F**•*d***r** is

Answer #1

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1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

Let f (x, y) = 100 sin(π(x−2y))/(1+x^2+y^2) . Find the
directional derivative of f 1+x^2+y^2 at the point (10, 6) in the
direction of: (a) u = 3 i − 2 j (b) v = −i + 4 j

Let C be a closed curve parametrized by r(t) = sin ti+cos tj
with 0 ≤ t ≤ 2π. Let F = yi − xj be a vector field.
(a) Evaluate the line integral xyds. C
(b) Find the circulation of F over C. (c) Find the flux of F
over C.

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

Let
F(x, y, z) =
(3x2 ln(6y2 + 2) +
8z3) i + (
12yx3
6y2 + 2
+ 7z) j + (24xz2 +
7y −
10π sin πz) k
and let r(t) = (t3
+ 1) i + (t2 +
2) j +
t3 k , 0 ≤ t ≤
1.
Evaluate
C
∫
C
F · dr .

y''(t) + 3y'(t) + 2y(t) = 0 if t < π 10 , sin(t) if t ≥ π ,
subject to y(0) = 5, y'(0) = 2

Evaluate C (y + 6 sin(x)) dx + (z2 + 2 cos(y)) dy + x3 dz where
C is the curve r(t) = sin(t), cos(t), sin(2t) , 0 ≤ t ≤ 2π. (Hint:
Observe that C lies on the surface z = 2xy.) C F · dr =

Let f(x,y) = xe^sin(x^2y+xy^2) /(x^2 + x^2y^2 + y^4)^3 . Compute
∂f ∂x (√2,0) pointwise.

4.Given F(x,y,z)=(cos(y))i+(sin(y))j+k, find divF and curlF at
P0(π/4,π,0) divF(P0)=? curlF(P0)= ?

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