Question

Use Green's first identity to show ∭|∇f|²dV = ∬(df/dn) dS.

Use Green's first identity to show ∭|∇f|²dV = ∬(df/dn) dS.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Answer: Let be a closed surface and be the region bounded by the surface . Let be the normal vector to the surface . Then, it follows from the definition of the directional derivative that

It also follows from the definition of the gradient and divergence that

Now, by Gauss' divergence theorem (Green's first identity), it follows that

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Use the divergence theorem to find the outward flux (F · n) dS S of the...
Use the divergence theorem to find the outward flux (F · n) dS S of the given vector field F. F = y2i + xz3j + (z − 1)2k; D the region bounded by the cylinder x2 + y2 = 25 and the planes z = 1, z = 6
Evaluate the surface integral    S F · dS for the given vector field F and...
Evaluate the surface integral    S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i − z j + y k S is the part of the sphere x2 + y2 + z2 = 4 in the first octant, with orientation toward the origin
Evaluate the surface integral S F · dS for the given vector field F and the...
Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i − z j + y k S is the part of the sphere x2 + y2 + z2 = 25 in the first octant, with orientation toward the origin
Evaluate the surface integral ∫∫S F · dS for the given vector field F and the...
Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i - z j + y k S is the part of the sphere x2 + y2 + z2 = 81 in the first octant, with orientation toward the origin.
Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate...
Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = x4i − x3z2j + 4xy2zk, S is the surface of the solid bounded by the cylinder x2 + y2 = 9 and the planes z = x + 4 and z = 0.
Use the Divergence Theorem ∬SF⋅dS=∭D∇⋅FdV ∬ S F ⋅ d S = ∭ D ∇ ⋅...
Use the Divergence Theorem ∬SF⋅dS=∭D∇⋅FdV ∬ S F ⋅ d S = ∭ D ∇ ⋅ F d V to find ∬SF⋅dS ∬ S F ⋅ d S where F(x,y,z)=3x2i+2y2j+2z2k F ( x , y , z ) = 3 x^2 i + 2 y^2 j + 2 z^2 k and S is the surface of the rectangular solid bounded by − 6 ≤ x ≤ 2 , − 6 ≤ y ≤ 3 , and − 4 ≤ z...
Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ∫ F...
Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ∫ F dx where F(x,y) =-yx^2i + xy^2j (lower bounds C) C consists of the circle x^2 + y^2 = 16 from (0,4) to(2√2, 2√2)and the line segments from (2√2, 2√2) to (0, 0) and from (0, 0) to (0,4)
Use Stokes' Theorem to evaluate    S curl F · dS. F(x, y, z) = x2...
Use Stokes' Theorem to evaluate    S curl F · dS. F(x, y, z) = x2 sin(z)i + y2j + xyk, S is the part of the paraboloid z = 1 − x2 − y2 that lies above the xy-plane, oriented upward.
Use Green's Theorem to evaluate the line integral ∫CF·dr along the given positively oriented curve. F(x,...
Use Green's Theorem to evaluate the line integral ∫CF·dr along the given positively oriented curve. F(x, y) = ‹ x2e-2x, x4 + 2x2y2 › C is the boundary of the region between the circles x2 + y2 = 1 and x2 + y2 = 25 in the 4th quadrant.
use stoke's theorem to find ∬ (curl F) * dS where F (x,y,z) = <y, 2x,...
use stoke's theorem to find ∬ (curl F) * dS where F (x,y,z) = <y, 2x, x+y+z> and and S is the upper half of the sphere x^2 + y^2 +z^2 =1, oriented outward
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT