Q. 47 Determine whether or not each of... [FREE SOLUTION]

Chapter 14: Q. 47 (page 1096)

Determine whether or not each of the vector fields in Exercises 41鈥�48 is conservative. If the vector field is conservative, find a potential function for the field.
$G (x, y, z) = (z 鈭� y) i 鈭� x y j + (x z + y) k$

Short Answer

Expert verified

Since, $\frac{鈭� G_{3}}{鈭� y} 鈮� \frac{鈭� G_{2}}{鈭� z}$ so the given vector is not conservative.

Step by step solution

Step 1. Given Information

We have to determine whether or not each of the vector fields in the given exercises is conservative. If the vector field is conservative, find a potential function for the field.

$G (x, y, z) = (z 鈭� y) i 鈭� x y j + (x z + y) k$

Step 2. Firstly finding the given vector field is conservative or not.

A vector field $G (x, y, z) = (G_{1} (x, y, z), G_{2} (x, y, z), G_{3} (x, y, z))$ is conservative if and only if localid="1650559661004" $\frac{鈭� G_{3}}{鈭� y} = \frac{鈭� G_{2}}{鈭� z}, \frac{鈭� G_{1}}{鈭� z} = \frac{鈭� G_{3}}{鈭� x}, \frac{鈭� G_{2}}{鈭� x} = \frac{鈭� G_{1}}{鈭� y}$

$\frac{鈭� G_{3}}{鈭� y} = \frac{鈭�}{鈭� y} (x z + y) \frac{鈭� G_{2}}{鈭� z} = \frac{鈭�}{鈭� z} (鈭� x y) \frac{鈭� G_{3}}{鈭� y} = \frac{鈭�}{鈭� y} x z + \frac{鈭�}{鈭� y} y \frac{鈭� G_{2}}{鈭� z} = 0 \frac{鈭� G_{3}}{鈭� y} = 0 + 1 \frac{鈭� G_{2}}{鈭� z} = 0 \frac{鈭� G_{3}}{鈭� y} = 1 \frac{鈭� G_{2}}{鈭� z} = 0$

Hence,data-custom-editor="chemistry" $\frac{鈭� G_{3}}{鈭� y} 鈮� \frac{鈭� G_{2}}{鈭� z}$

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91影视

Determine whether or not each of the vector fields in Exercises 41鈥�48 is conservative. If the vector field is conservative, find a potential function for the field.
$G (x, y, z) = (z 鈭� y) i 鈭� x y j + (x z + y) k$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly finding the given vector field is conservative or not.

One App. One Place for Learning.

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91影视

Determine whether or not each of the vector fields in Exercises 41鈥�48 is conservative. If the vector field is conservative, find a potential function for the field.G(x,y,z)=(z鈭�y)i鈭�xyj+(xz+y)k

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly finding the given vector field is conservative or not.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Probability and Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Determine whether or not each of the vector fields in Exercises 41鈥�48 is conservative. If the vector field is conservative, find a potential function for the field.
$G (x, y, z) = (z 鈭� y) i 鈭� x y j + (x z + y) k$