Chapter 6: Q26MP (page 338)

$\iint V \times ndσ$ over the entire surface of the cube in the first octant with three faces in the three coordinate planes and the other three faces intersecting at $(2, 2, 2)$ , where $V = (2 - y) i + xzj + xyzk$ .

Short Answer

Expert verified

The Solution is $\iint_{\partial T} (\nabla \times V) \times ndσ = 0 .$

Step by step solution

Given Information.

The given information is, $V = (2 - y) i + xzj + xyzk$

Definition of Divergence Theorem.

The divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed.

Find the solution.

Use the divergence theorem $∭_{T} \nabla \times VdT = \iint_{\partial T} V \times ndσ$ , where is the surface area that encloses the volume $T$ .

$\nabla \times (\nabla \times V) = 0$

But the divergence of any curve is zero. Use the triplet scale product.

$\iint_{\partial T} (\nabla \times V) \times ndσ = \iint_{J} \nabla \times (\nabla \times V) dT$

$= 0$

Hence, the solution is $\iint_{\partial T} (\nabla \times V) \times ndσ = 0 .$

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$\iint V \times ndσ$ over the entire surface of the cube in the first octant with three faces in the three coordinate planes and the other three faces intersecting at $(2, 2, 2)$ , where $V = (2 - y) i + xzj + xyzk$ .

Short Answer

Step by step solution

Given Information.

Definition of Divergence Theorem.

Find the solution.

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∬V×ndσover the entire surface of the cube in the first octant with three faces in the three coordinate planes and the other three faces intersecting at (2,2,2), where V=(2−y)i+xzj+xyzk.

Short Answer

Step by step solution

Given Information.

Definition of Divergence Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Fields in Physics

Solid State Physics

Force

Wave Optics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

$\iint V \times ndσ$ over the entire surface of the cube in the first octant with three faces in the three coordinate planes and the other three faces intersecting at $(2, 2, 2)$ , where $V = (2 - y) i + xzj + xyzk$ .