Chapter 6: Q9P (page 334)

$\iint V \times ndσ$ over the entire surface of the volume in the first octant bounded by $x^{2} + y^{2} + z^{2} = 16$ and the coordinate planes, where $V = (x + x^{2} - y^{2}) i + (2 xyz - 2 xy) j - {xz}^{2} k$

Short Answer

Expert verified

The solution derived is $I = \frac{32}{3} π$

Step by step solution

Given Information.

The given equation is $V = (x + x^{2} - y^{2}) i + (2 xyz - 2 xy) j - {xz}^{2} k$

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 its magnitude.

Apply divergence theorem.

Apply the divergence theorem and the fact that $\nabla \times V = 1 + 2 x + 2 xz - 2 x - 2 xz = 1$ then the integral becomes as shown below.

$I = ∭_{V} dτ$

$= \frac{1}{8} \frac{4}{3} π 4^{3}$

$= \frac{32}{3} π$

Note that only $\frac{1}{8}$ of the volume of the sphere is considered because only the first octant is operated.

Hence, the solution derived is $I = \frac{32}{3} π$ .

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$\iint V \times ndσ$ over the entire surface of the volume in the first octant bounded by $x^{2} + y^{2} + z^{2} = 16$ and the coordinate planes, where $V = (x + x^{2} - y^{2}) i + (2 xyz - 2 xy) j - {xz}^{2} k$

Short Answer

Step by step solution

Given Information.

Definition of vector.

Apply divergence theorem.

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∬V×ndσover the entire surface of the volume in the first octant bounded byx2+y2+z2=16and the coordinate planes, whereV=x+x2−y2i+(2xyz−2xy)j−xz2k

Short Answer

Step by step solution

Given Information.

Definition of vector.

Apply divergence theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Fundamentals of Physics

Space Physics

Electromagnetism

Physical Quantities And Units

Measurements

Study anywhere. Anytime. Across all devices.

$\iint V \times ndσ$ over the entire surface of the volume in the first octant bounded by $x^{2} + y^{2} + z^{2} = 16$ and the coordinate planes, where $V = (x + x^{2} - y^{2}) i + (2 xyz - 2 xy) j - {xz}^{2} k$