Q. 35 Show that the vector fields in E... [FREE SOLUTION]

Chapter 14: Q. 35 (page 1096)

Show that the vector fields in Exercises 33鈥�40 are not conservative.
$G (x, y) = \frac{1}{x^{2} + y} i + \frac{y}{x} j$

Short Answer

Expert verified

The vector fields is not conservative because $\frac{鈭� G_{1}}{鈭� y} 鈮� \frac{鈭� G_{2}}{鈭� x}$ .

Step by step solution

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
$G (x, y) = \frac{1}{x^{2} + y} i + \frac{y}{x} j$

Step 2. A vector field G(x,y)=(G1(x,y),G2(x,y)) is not conservative if and only if ∂G1∂y≠∂G2∂x.

For the vector field

$\frac{鈭� F_{1}}{鈭� y} = \frac{鈭�}{鈭� y} \frac{1}{x^{2} + y} \frac{鈭� F_{1}}{鈭� y} = \frac{鈭�}{鈭� y} {(x^{2} + y)}^{- 1} \frac{鈭� F_{1}}{鈭� y} = - 1 {(x^{2} + y)}^{- 2} \frac{鈭� F_{1}}{鈭� y} = - {(x^{2} + y)}^{- 2}$

Step 3. Now finding ∂F2∂y

$\frac{鈭� F_{2}}{鈭� x} = \frac{鈭�}{鈭� x} (\frac{y}{x}) \frac{鈭� F_{2}}{鈭� x} = y \frac{鈭�}{鈭� x} (\frac{1}{x}) \frac{鈭� F_{2}}{鈭� x} = y \frac{鈭�}{鈭� x} x^{- 1} \frac{鈭� F_{2}}{鈭� x} = - y x^{- 2}$

Hence, $\frac{鈭� G_{1}}{鈭� y} 鈮� \frac{鈭� G_{2}}{鈭� x} .$

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Show that the vector fields in Exercises 33鈥�40 are not conservative.
$G (x, y) = \frac{1}{x^{2} + y} i + \frac{y}{x} j$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. A vector field G(x,y)=(G1(x,y),G2(x,y)) is not conservative if and only if ∂G1∂y≠∂G2∂x.

Step 3. Now finding ∂F2∂y

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91影视

Show that the vector fields in Exercises 33鈥�40 are not conservative.G(x,y)=1x2+yi+yxj

Short Answer

Step by step solution

Step 1. Given Information

Step 2. A vector field G(x,y)=(G1(x,y),G2(x,y)) is not conservative if and only if ∂G1∂y≠∂G2∂x.

Step 3. Now finding ∂F2∂y

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Calculus

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Show that the vector fields in Exercises 33鈥�40 are not conservative.
$G (x, y) = \frac{1}{x^{2} + y} i + \frac{y}{x} j$