Let⇀F(x,y) =xi+yj/(e^(x^2+y^2))−1. Let C be a positively oriented simple closed path that encloses the origin. (a)...

Let⇀F(x,y) =xi+yj/(e^(x^2+y^2))−1. Let C be a positively oriented simple closed path that encloses the origin.

(a) Show that∫F·Tds= 0.

(b) Is it true that ∫F·Tds= 0 for any positively oriented simple closed path that does not pass through or enclose the origin? Justify your response completely.

Solutions

Expert Solution

Dear student,

The problem is completely solved below.

Feel free to ask doubts in the comments section..

Justification:

For stokes theorem only you need to think is where the curve is CLOSED or not. Then only you will be able to form a surface integral which is not defined without a closed boundary. Here This theorem is not based on any reference point like origin or some other point.Also you MUST ensure the that is integrable (piece wise integrability is also accepted.)These are the only conitinous that stokes theorem demands.

The above curve satisfied integrability condition and closed.So it has freedom to get the theorem applied on it.

According to definition,

Here curl is zero irrespective of any condition.So you can always choose a closed path that may not pass through origin or may not contain origin in it.

If you feel still confused.Feel free to comment your doubt.

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

1. F=xi+yj+ (xz+yz)k, and Sis the part ofx+y+z= 1 in the first octant, oriented upwards. Compute∫∫Scurl(F)·dSdirectly....

1. F=xi+yj+ (xz+yz)k, and Sis the part ofx+y+z= 1 in the first octant, oriented upwards. Compute∫∫Scurl(F)·dSdirectly. 2. Compute∫CF·drdirectly (3 curves to parameterize).

Evaluate the line integral, where C is the given curve. ∫CF(x,y,z)⋅dr where F(x,y,z)=xi+yj+ysin(z+1)k and C consists...

Evaluate the line integral, where C is the given curve. ∫CF(x,y,z)⋅dr where F(x,y,z)=xi+yj+ysin(z+1)k and C consists of the line segment from (2,4,-1) to (1,-1,3).

Let C be the closed path that travels from (0, 0) to (1, 1) along y...

Let C be the closed path that travels from (0, 0) to (1, 1) along y = x^2 , then from (1, 1) to (0, 2) along y = 2 − x^ 2 , and finally in a straight line from (0, 2) to (0, 0). Evaluate Z C e ^3−x √ 3 − x ) dx + (5x − y √ y^2 + 2) dy

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x <...

Let X and Y have joint PDF f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y < infinity with parameters λ > 0 and μ > 0 a) Find c such that this is a PDF. b) Show that X and Y are Independent c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in exponential form d) Find the marginal distribution of Y, f(y) e) Find E(Y)

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y...

Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y + 1, be the joint pdf of X and Y. (a) (3 pts) Find c and sketch the region for which f (x, y) > 0. (b) (3 pts) Find fX(x), the marginal pdf of X. (c) (3 pts) Find fY(y), the marginal pdf of Y. (d) (3 pts) Find P(X ≤ 3 − Y). (e) (4 pts) E(X) and Var(X). (f) (4 pts) E(Y)...

1. Find absolute max and min of f(x,y)= x^2- xy + y^2 +1 on the closed...

1. Find absolute max and min of f(x,y)= x^2- xy + y^2 +1 on the closed triangular plate in the first quadrant x=0, y=4, y=x 2. Given position of a particle by π (t)= Cos2ti + 3 sin2ti, Find the particle velocity and acceleration at t=0

Let the joint pmf of X and Y be defined by f (x, y) = c(x...

Let the joint pmf of X and Y be defined by f (x, y) = c(x + y), x =0, 1, 2, y = 0, 1, with y ≤ x. 1. Are X and Y independent or dependent? Why or why not? 2. Find g(x | y) and draw a figure depicting the conditional pmfs for y =0 and 1. 3. Find h(y | x) and draw a figure depicting the conditional pmfs for x = 0, 1 and2. 4....

Let the joint pmf of X and Y be defined by f (x, y) = c(x...

Let the joint pmf of X and Y be defined by f (x, y) = c(x + y), x =0, 1, 2, y = 0, 1, with y ≤ x. 1. Find g(x | y) and draw a figure depicting the conditional pmfs for y =0 and 1. 2. Find h(y | x) and draw a figure depicting the conditional pmfs for x = 0, 1 and2. 3. Find P(0 < X <2 |Y = 0), P(X ≤ 2 |Y...

Let f ( x , y ) = x^ 2 + y ^3 + sin ⁡...

Let f ( x , y ) = x^ 2 + y ^3 + sin ⁡ ( x ^2 + y ^3 ). Determine the line integral of f ( x , y ) with respect to arc length over the unit circle centered at the origin (0, 0).

Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x...

Let the random variable and have the joint pmf X Y f(x,y) = {x(y)^2}/c where x = 1, 2, 3 ; y = 1, 2, x+y<= 4, that is (x,y) are {(1,1), (1,2), (2,1), (2,2), 3,1)} (a) Find . c > 0 (b) Find . μX (c) Find . μY (d) Find . σ2 X (e) Find . σ2 Y (f) Find Cov . (X,Y ) (g) Find p , Corr . (x,y) (h) Are and X and Y independent

Subjects