Q. 65 Prove that if F(x,聽y,聽z) is a ... [FREE SOLUTION]

Chapter 14: Q. 65 (page 1108)

Prove that if $F (x, y, z)$ is a conservative vector field, then line integrals of $F (x, y, z)$ are independent of path.

Short Answer

Expert verified

Hence, we prove that $F (x, y, z)$ are independent of path.

Step by step solution

Step 1. Given Information

Prove that if $F (x, y, z)$ is a conservative vector field, then line integrals of $F (x, y, z)$ are independent of path.

Step 2. As we know that F is a conservative vector field.

$鈭� f 鈰� d r = {鈭�}_{a}^{b} 鈭� f (r (t)) 鈰� r' (t) d t 鈭� f 鈰� d r = {鈭�}_{a}^{b} (\frac{鈭� f}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� f}{鈭� y} 路 \frac{d y}{d t} + \frac{鈭� f}{鈭� z} 路 \frac{d z}{d t}) d t$

Now, at this point we can use the Chain Rule to simplify the integrand as follows,

$鈭� f 鈰� d r = {鈭�}_{a}^{b} (\frac{鈭� f}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� f}{鈭� y} 路 \frac{d y}{d t} + \frac{鈭� f}{鈭� z} 路 \frac{d z}{d t}) d t 鈭� f 鈰� d r = {鈭�}_{a}^{b} \frac{d}{d t} [f (r (t))] d t$

To finish this off we just need to use the Fundamental Theorem of Calculus for single integrals.

$鈭� f 鈰� d r = f (r (b)) - f (r (a))$

The integral depends only on the endpoints of C and not on C itself.

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91影视

Prove that if $F (x, y, z)$ is a conservative vector field, then line integrals of $F (x, y, z)$ are independent of path.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. As we know that F is a conservative vector field.

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91影视

Prove that if F(x,y,z) is a conservative vector field, then line integrals of F(x,y,z) are independent of path.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. As we know that F is a conservative vector field.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Prove that if $F (x, y, z)$ is a conservative vector field, then line integrals of $F (x, y, z)$ are independent of path.