Question

How can we prove $\langle(Ψ\cdot∇)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}+\frac12\langle(∇\cdotΨ)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}=0$ for $Φ,Ψ\in H_0^1(Λ,ℝ^d)$?

score 46 · Answer 1 · 2026-03-26 16:10:18

46

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How can we prove $\langle(Ψ\cdot∇)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}+\frac12\langle(∇\cdotΨ)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}=0$ for $Φ,Ψ\in H_0^1(Λ,ℝ^d)$?

Published on 26 Mar 2026 - 16:10

score 356 · Answer 2 · 2026-03-27 08:42:00

356

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Calculate the divergence of a vector field without the definition

Published on 27 Mar 2026 - 8:42

score 700 · Answer 3 · 2026-04-01 03:03:30

700

Views

Why is there a difference between coordinate variables and vector components when transforming coordinates?

Published on 01 Apr 2026 - 3:03

score 1.6k · Answer 4 · 2026-03-27 14:56:46

Given a vector field $\vec{F} = zy\hat{x} + zx\hat{y} + xy\hat{z}$ evaluate the integral $\vec{F} \cdot d\vec{l}$ from $(0,0,0)$ to $(1,2,3)$.

score 326 · Answer 5 · 2026-04-02 11:56:00

326

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Differential forms and vector analysis

Published on 02 Apr 2026 - 11:56

score 292 · Answer 6 · 2017-02-24 10:51:15

292

Views

Is the Laplace operator w.r.t. all coordinates or not?

Published on 24 Feb 2017 - 10:51

score 484 · Answer 7 · 2017-02-25 10:47:14

484

Views

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Published on 25 Feb 2017 - 10:47

score 172 · Answer 8 · 2017-02-25 18:45:49

172

Views

Prove that the gradient of a 2 variable function behave like a vector under 2-D rotations.

Published on 25 Feb 2017 - 18:45

score 26 · Answer 9 · 2017-02-25 21:26:52

26

Views

Solve for $\epsilon(k)$ then show $\bigtriangledown \epsilon$ is normal to $\epsilon_n(k)$

Published on 25 Feb 2017 - 21:26

score 293 · Answer 10 · 2026-03-26 10:45:01

293

Views

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score 581 · Answer 11 · 2026-03-26 22:17:22

581

Views

proof of a fluid dynamics identity using index notation

Published on 26 Mar 2026 - 22:17

score 30 · Answer 12 · 2017-02-27 02:12:30

30

Views

Unclear if this del vector identity is true

Published on 27 Feb 2017 - 2:12

score 94 · Answer 13 · 2017-02-28 20:30:45

94

Views

What are the methods to manipulate a gradient?

Published on 28 Feb 2017 - 20:30

score 62 · Answer 14 · 2026-03-27 11:47:44

62

Views

What is the difference between the line integrals $\oint_b\,ds$, $\oint_b\,dx$, and $\oint_b\,x\,ds$?

Published on 27 Mar 2026 - 11:47

score 161 · Answer 15 · 2026-03-27 11:46:39

161

Views

Line integral of an exact expression

Published on 27 Mar 2026 - 11:46

List Question

How can we prove $\langle(Ψ\cdot∇)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}+\frac12\langle(∇\cdotΨ)Φ,Φ\rangle_{L^2(Λ,\:ℝ^d)}=0$ for $Φ,Ψ\in H_0^1(Λ,ℝ^d)$?

Calculate the divergence of a vector field without the definition

Why is there a difference between coordinate variables and vector components when transforming coordinates?

Given a vector field $\vec{F} = zy\hat{x} + zx\hat{y} + xy\hat{z}$ evaluate the integral $\vec{F} \cdot d\vec{l}$ from $(0,0,0)$ to $(1,2,3)$.

Differential forms and vector analysis

Is the Laplace operator w.r.t. all coordinates or not?

What is a conservative field?

Prove that the gradient of a 2 variable function behave like a vector under 2-D rotations.

Solve for $\epsilon(k)$ then show $\bigtriangledown \epsilon$ is normal to $\epsilon_n(k)$

Gradient of a softmax layer cases issue

proof of a fluid dynamics identity using index notation

Unclear if this del vector identity is true

What are the methods to manipulate a gradient?

What is the difference between the line integrals $\oint_b\,ds$, $\oint_b\,dx$, and $\oint_b\,x\,ds$?

Line integral of an exact expression

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