Chapter 12: Q. 67 (page 945)

$Recall that the volume, V, of a cylinder is given by V (r, h) = π r^{2} h . Find \frac{\partial V}{\partial r} and \frac{\partial V}{\partial h} . What are the physical interpretations of these partial derivatives ?$

Short Answer

Expert verified

$\frac{\partial V}{\partial r} = 2 π r h This tells us how fast the volume changes with a unit change in the radius . \frac{\partial V}{\partial h} = π r^{2} This tells us how fast the volume changes with a unit change in the height .$

Step by step solution

Step 1. Given information

The volume of a cylinder is given by,

V (r, h) = π r^{2} h

Step 2. Finding the partial derivatives and physical interpretation of them

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.

$Recall that the volume, V, of a cylinder is given by V (r, h) = π r^{2} h . Find \frac{\partial V}{\partial r} and \frac{\partial V}{\partial h} . What are the physical interpretations of these partial derivatives ?$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the partial derivatives and physical interpretation of them

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Statistics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Recallthatthevolume,V,ofacylinderisgivenbyV(r,h)=πr2h.Find∂V∂rand∂V∂h.Whatarethephysicalinterpretationsofthesepartialderivatives?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the partial derivatives and physical interpretation of them

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Mechanics Maths

Statistics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

$Recall that the volume, V, of a cylinder is given by V (r, h) = π r^{2} h . Find \frac{\partial V}{\partial r} and \frac{\partial V}{\partial h} . What are the physical interpretations of these partial derivatives ?$