Chapter 6: Q5P (page 334)

Question: $\int \int r \cdot ndσ$ over the surface in Problem 4, where r = ix + jy + kz. Hint: See Problem 10.9.

Short Answer

Expert verified

The solution is derived to be $\int \int_{indind plan} r \cdot ndσ = 36$ .

Step by step solution

Given Information.

It has been asked to use problem 4.and r = ix + jy + kz.

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines it magnitude.

Apply Gauss theorem.

Apply Gauss' theorem, and the fact that $\nabla \cdot r = 3$

$\int \int_{indind plan} r \cdot ndσ = \oint_{v} \nabla \cdot rdt - \int \int_{yz} r \cdot ndσ - \int \int_{xz} r \cdot ndσ - \int \int_{xy} r \cdot ndσ$

The first term is simply reduced to be $l_{1} = 3 V$ where V the volume of the entire shape is, calculate it,

$V = \frac{1}{2} \int_{0}^{3} (6 - 2 z) (4 - \frac{4}{3} z) dz = 12 l_{1} = 36$

The second term is : $l_{2} = - \int \int xdσ but x = 0, then l_{2} = 0$

The third term is: $l_{3} = - \int \int xdσ but y = 0, then l_{3} = 0$

The fourth term is: $l_{4} = - \int \int xdσ but z = 0, then l_{4} = 0$

Hence, the solution is derived to be $\int \int_{indind plan} r \cdot ndσ = 36$

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Question: $\int \int r \cdot ndσ$ over the surface in Problem 4, where r = ix + jy + kz. Hint: See Problem 10.9.

Short Answer

Step by step solution

Given Information.

Definition of vector.

Apply Gauss theorem.

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Question: ∫∫r·ndσ over the surface in Problem 4, where r = ix + jy + kz. Hint: See Problem 10.9.

Short Answer

Step by step solution

Given Information.

Definition of vector.

Apply Gauss theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Work Energy and Power

Oscillations

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Study anywhere. Anytime. Across all devices.

Question: $\int \int r \cdot ndσ$ over the surface in Problem 4, where r = ix + jy + kz. Hint: See Problem 10.9.