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.

$\nabla f \cdot d r = \int_{a}^{b} \nabla f (r (t)) \cdot r' (t) d t \nabla f \cdot d r = \int_{a}^{b} (\frac{\partial f}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial f}{\partial y} \cdot \frac{d y}{d t} + \frac{\partial f}{\partial z} \cdot \frac{d z}{d t}) d t$

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

$\nabla f \cdot d r = \int_{a}^{b} (\frac{\partial f}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial f}{\partial y} \cdot \frac{d y}{d t} + \frac{\partial f}{\partial z} \cdot \frac{d z}{d t}) d t \nabla f \cdot d r = \int_{a}^{b} \frac{d}{d t} [f (r (t))] d t$

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

$\nabla f \cdot d r = f (r (b)) - f (r (a))$

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

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Geometry

Discrete Mathematics

Decision Maths

Mechanics Maths

Calculus

Statistics

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.

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

Geometry

Discrete Mathematics

Decision Maths

Mechanics Maths

Calculus

Statistics

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.