This page covers one application of gradients and directional derivatives, the temperature gradient.
Recommended Books on Amazon (affiliate links) | ||
---|---|---|
![]() |
![]() |
![]() |
Temperature
When the function \(f\) is a temperature, the gradient can be interpreted as the change in temperature.
Practice
If we have a material where the heat flow (from hot to cold) is given by \( T = x^3 - 3xy^2 \), determine the direction of maximum decrease of temperature at the point \( (1,2) \).
Problem Statement
If we have a material where the heat flow (from hot to cold) is given by \( T = x^3 - 3xy^2 \), determine the direction of maximum decrease of temperature at the point \( (1,2) \).
Solution
video by Dr Chris Tisdell |
---|
Log in to rate this practice problem and to see it's current rating. |
---|
The temperature, in degrees Celsius, on the surface of a metal is given by
\( T(x,y) = 20-4x^2-y^2 \), where \(x\) and \(y\) are measured in centimeters.
(a) In what direction, from \((2,-3)\), does the temperature increase most rapidly? What is this rate of increase?
(b) A heat-seeking particle is located at \((2,-3)\) on the plate. Find the path of the particle as it continuously moves in the direction of maximum temperature increase.
Problem Statement
The temperature, in degrees Celsius, on the surface of a metal is given by
\( T(x,y) = 20-4x^2-y^2 \), where \(x\) and \(y\) are measured in centimeters.
(a) In what direction, from \((2,-3)\), does the temperature increase most rapidly? What is this rate of increase?
(b) A heat-seeking particle is located at \((2,-3)\) on the plate. Find the path of the particle as it continuously moves in the direction of maximum temperature increase.
Solution
video by Thomas Wernau |
---|
Log in to rate this practice problem and to see it's current rating. |
---|
Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
external links you may find helpful |
---|
Better Explained: Vector Calculus: Understanding the Gradient |
To bookmark this page and practice problems, log in to your account or set up a free account.
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
| |
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. |
---|
Support 17Calculus on Patreon |
|
---|