For each of the following, i) Determine if the vector field F is a conservative

ID: 2853295 • Letter: F

Question

For each of the following,

i) Determine if the vector field F is a conservative vector field. If F is conservative, find a function f such that (f) = F.

ii) Evaluate the line integral ! C F·dr along the given curve C . Be sure to justify your answer fully.

(a) F(x, y)=(x^2 4y^2)i + (2x 3)j. C is the line segment from (0, 1) to (2, 0).

(b) F(x, y, z)=2xy cos(z)i + x^2 cos(z)j (x^2)y sin(z)k C is given by r(t) = cos(t)i + ( t/2 + sin(t))j + tk where 0 t 2.

(c) F(x, y) = yi + xj C consists of the curve y = ln(x+1)/ln(2) from (0, 0) to (1, 1) followed by the curve x = ln(y+1)/ln(2) from (1, 1) to (0, 0).

ii) Evaluate the line integral ! C F·dr along the given curve

(a) F(x, y) = (x^2 4y^2)i + (2x 3)j. C is the line segment from (0, 1) to (2, 0).

here The direction of the line is ( 2- 0, 0 -1) = ( 2, -1)

So its equation is x = 2t, y = -1 - t

r(t) = < 2t, -1 -t >

r'(t) = < 2, -1 >

Therefore , required line integral is

l ! C F·dr = [ (x^2 4y^2)i + (2x 3)j ] . [ 2i ,-j ] drdy

= Integral from t = 0 to t = 1 [ (2(2t^2 8(-1 -t)^(2) - (2*(2t) 3) ] dt

= Integral from t = 0 to t = 1 [ 4t^(2) - 8 ( 1 + t^2 +2t ) - 4t +3 ] dt

= " [ 4t^(2) - 8 - 8t^2 - 16t ) - 4t +3 ] dt

= " [ - 4t^(2) - 5 - 20t ) ] dt

= - Integral from t = 0 to t = 1 [ 4t^(2) + 20t + 5 ] dt

Now, on integrating this , we get

= -19 /3

b) To Evaluate the line integral ! C F·dr along the given Curve

We have , F(x,y,z) = 2xy cos(z) i + x^(2) y sinz k

where C is given by r(t) = cos(t)i + ( t/2 + sin(t))j + tk where 0 t 2.

Also , r'(t) = -sinti + (1/2 + cost )j + k

Now, line integral C F·dr =

= Integral from t = 0 to t =2pi [ 2xy cos(z) + x^(2) y sinz ] [ -sinti + (1/2 + cost )j + k ] dt

= Integral from t = 0 to t =2pi [ - 2cos(t)[ t/2 + sint ] costsint + cos^(2)t (t/2 + sint)( sint ) ]dt

On integrating this , we get

= 0

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