Can someone please help me with these 2 problems? I appreciate your time. Use st
ID: 2862311 • Letter: C
Question
Can someone please help me with these 2 problems? I appreciate your time.
Use stokes Theorem to evaluate integral_c F. dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = e^-x_i + e^-y_j + e^-z_k, C is the boundary of the part of the plane 6x + y + 6z = 6 in the first octant.________ Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux. F(x, y, z) = 3xi + xyj + 5xzk, E is the cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z = 0, and z = 2.________Explanation / Answer
ans-12
Curl F= (0,0, e^x)
INT_C F.dr = INT_S curlF.<n> dS where S is the part of the plane in the first Octant .
Hint ; dS projected over XY plane is <n<dS .k = dA , so dS= I N I dA/ I N. kI ,
because <n> = N / I N I -----mode N
Now, <n> dS = N/INI (INI dA/ int.kI ) so finally , <n> dS= N/IN.kI dA
INT_S curlF dS= INT_A Curl F . N/I N.kI dA
N of the surface ( Plane ) is N=(6,1,6) and thus, N.k= 6
INT :C F,dr = INT_A (0,0,e^x) . (1/6) (6,1,6) dA
=INT_A e^x dA
At z=0 , the boundary on the XY plane is the line 6x+y=6 , so the intercepts ( X,Y) are
(1,0) ( X axis)
(0,6) ( Y Axis)
INT _Ae^x dydx = INT y e^x dx
0<y<(6-6x)
= INT (6-6x)e^x dx
0<x<1
= 6INT e^x dx -6INT xe^x dx
= 6e^x -6 ( xe^x -e^x)
=12e^x-6xe^x for 0<x<1 , so
= (12e-6e)- (12)
=6e-12
=6(e- 2) -------------answer