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The heat flow vector field for conducting objects is F=−k∇T​, where T(x,y,z) is the temperature in the object and k is a constant that depends on the material. Compute the outward flux of F across the given surface S for the given temperature distribution. Assume k=1. T(x,y,z,)=−7ln( x2 +y2 + z2)S is the sphere x2+y2+z2=a2

Question
The heat flow vector field for conducting objects is F=−k∇T​, where T(x,y,z) is the temperature in the object and k is a constant that depends on the material. Compute the outward flux of F across the given surface S for the given temperature distribution. Assume k=1.
 
T(x,y,z,)=−7ln( x2 +y2 + z2)
S is the sphere x2+y2+z2=a2
check_circleAnswer
Step 1

Assume k = 1.

Then the vector field becomes, F = −∇T.

Find ∇T as follows.

от от от
УТ 3
ах ду д
-7
-7
-7
2у.
x* + y* +z*) ^ (x2? + y? +)
2х,
2z
+y?+*)
-14z
-14х
-14 у
(x +y*+
z*)(x* +y? + z*) (* +y? +z°) }
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от от от УТ 3 ах ду д -7 -7 -7 2у. x* + y* +z*) ^ (x2? + y? +) 2х, 2z +y?+*) -14z -14х -14 у (x +y*+ z*)(x* +y? + z*) (* +y? +z°) }

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Step 2

The parametric description of the sphere of radius a, centered at the origin is given by,

r(u,v)(asiucosv, asin usinv,acosu), 0<usT, 0sv27
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r(u,v)(asiucosv, asin usinv,acosu), 0<usT, 0sv27

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Step 3

Take the partial derivative of the parameterization with...

(asin ucosv)(asin usin v), (acosu)}
\au
' ôu
(a cosucos v, acosusin v,-asinu)
(asinucosv)(asin usin v),(acosu)
'av
\av
=(-asinusin v,asinucosv,0)
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(asin ucosv)(asin usin v), (acosu)} \au ' ôu (a cosucos v, acosusin v,-asinu) (asinucosv)(asin usin v),(acosu) 'av \av =(-asinusin v,asinucosv,0)

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