Use Stokes' Theorem to evaluate int_C F*dr where C is oriented counterclockwise as viewed above.F(x,y,z)=xy i + 3zj+5yk, C is the curve of intersection of the plane x+z=10 and the cylinder x^2+y^2=9.

EunoR

EunoR

Answered question

2021-01-06

Use Stokes' Theorem to evaluate CFdr where C is oriented counterclockwise as viewed above.
F(x,y,z)=xyi+3zj+5yk, C is the curve of intersection of the plane x+z=10 and the cylinder x2+y2=9.

Answer & Explanation

Jaylen Fountain

Jaylen Fountain

Skilled2021-01-07Added 169 answers

Step 1
Here use the stokes theorem to find the integral of the function.
And given that the function F(x,y,z)=xyi+3zj+5yk
The curve of intersection of the plane x+z=10 and the cylinder x2+y2=9
Step 2
The Stokes theorem is given as:
CFdr=ScurlFds
if F=Pi^+Qj^+Rk^
Then CurlF=(RyQz)i^+(PzRx)j^+(QxPy)k^
Given F=xyi^+3zj^+5yk^
Therefore,
Curl F=(53)i^+(00)j^+(0x)k^=2i^xk^
Step 3
Let S be the path of the plane x+z=10 inside the cylinder x2+y2=9
If the surface S is of the form z=g(x,y) and is oriented upwards,
So, by the formula:
SFdS=DP(gxQgy)+RdA
Here S is the part of the plane z=10x
And D is region inside the circle x2+y2=0,z=0
SCurlFds=D(2x)dA
Step 4
Solve further:
SFdS==02π03(2r2cos0)rdrd0
=02π[032rdr03r3cos(0)dr]d0
=02π[r22+r44cos0]03d0

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