Chapter 1: Q15P (page 18)

Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Short Answer

Expert verified

(a) The divergence is 0.

(b) The divergence is $^{3 x + y + 2 z}$ .

Step by step solution

Write the expression for the divergence of the function.

Determine the divergence of the function.

$\nabla . v = (\hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z}) (v_{x} \hat{x} + v_{y} \hat{y} + v_{z} \hat{z}) = \frac{\partial v_{x}}{\partial x} + \hat{y} \frac{\partial v_{y}}{\partial y} + \hat{z} \frac{\partial v_{z}}{\partial z}$

Solve for the divergence of part (a).

Consider the given vector is,

$v_{a} = x^{2} \hat{x} + 3 x z^{2} y - 2 x z \hat{z}$

Solve for the divergence.

$\nabla . v_{a} = \frac{\partial}{\partial x} x^{2} + \frac{\partial}{\partial y} 3 x z^{2} - \frac{\partial}{\partial z} 2 x z = 2 x + 0 - 2 x = 0$

Therefore, the divergence is 0.

Solve for the divergence of part (b).

Consider the given vector is,

$v_{b} = x y \hat{x} + 2 y z^{2} y + 3 z x \hat{z}$

Solve for the divergence.

$\nabla . v_{a} = \frac{\partial}{\partial x} x y + \frac{\partial}{\partial y} 2 y z - \frac{\partial}{\partial z} x z = 3 x + y + 2 z$

Therefore, the divergence is $^{3 x + y + 2 z}$

Solve for the divergence of part (c).

Consider the given vector is,

$v_{a} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Solve for the divergence.

$\nabla . v_{a} = \frac{\partial}{\partial x} y^{2} + \frac{\partial}{\partial y} (2 x y + z^{2}) + \frac{\partial}{\partial z} 2 y z = 0 + (2 x + 0) + 2 y = 2 (x + y)$

Therefore, the divergence is $^{2 (x + y)}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Short Answer

Step by step solution

Write the expression for the divergence of the function.

Solve for the divergence of part (a).

Solve for the divergence of part (b).

Solve for the divergence of part (c).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Oscillations

Electricity

Quantum Physics

Geometrical and Physical Optics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

Calculate the divergence of the following vector functions:(a)va=x2x^+3xz2y^-2xzz^(b)vb=xyx^+2yzy^+3zxz^(c)vc=y2x^+(2xy+z2)+2yzz^

Short Answer

Step by step solution

Write the expression for the divergence of the function.

Solve for the divergence of part (a).

Solve for the divergence of part (b).

Solve for the divergence of part (c).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Oscillations

Electricity

Quantum Physics

Geometrical and Physical Optics

Electricity and Magnetism

Study anywhere. Anytime. Across all devices.

Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$