Temperature Gradient Questions

In summary, we are trying to find the direction A in which the rate of change of the temperature function T(x,y,x)=2xy - yz at P(1,-1,1) is -3ºC/ft. After finding the gradient and setting it equal to the given rate, the equation cannot be solved for. In the second problem, the temperature function is given by T(x,y,z) = 2x^2 -xyz and a particle's position is given by x=2t^2, y=3t, z=-t^2. The rate of change of temperature per meter and per second is calculated using the chain rule and substituting in values for x, y, and z.
  • #1
Mindscrape
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Is there a direction A in which the rate of change of the temperature function [tex]T(x,y,x)=2xy - yz[/tex] at P(1,-1,1) is -3ºC/ft? Give reasons for your answer.

For this problem I found the gradient of at the point P. So
[tex]\nabla f = 2y|_p \mathbf{i} + 2x-z|_p \mathbf{j} - y|_p \mathbf{k}[/tex]
which I then found was
[tex] \nabla f = -2 \mathbf{i} + 1 \mathbf{j} + 1 \mathbf{k} [/tex]
Then we want to know if the gradient, in the direction of A will be 3ºC, so
[tex] \nabla f \cdot \frac{A}{|A|} = \frac{3º}{ft}[/tex]
which means that we want an x, y, z, such that
[tex]\frac{-2x +y +z}{\sqrt{x^2 + y^2 + z^2}} = \frac{3º}{ft}[/tex]
So I believe this is right so far, but I don't think this eqn can be solved for.

Here is the other one:
The Celsius temperature in a region in space is given by [tex]T(x,y,z) = 2x^2 -xyz[/tex]. A particle is moving in this region and its position at time t is given by [tex]x=2t^2[/tex], [tex]y=3t[/tex], [tex]z=-t^2[/tex], where time is measured in seconds and distanace in meters.
a) How fast is the temp experienced by the particle changing in ºC/m when the particle is at the point P(8,6,-4)?
b) How fast is the temp experienced by the particle changing in ºC/sec at P?

So, using the chain rule
[tex] \frac{dT}{dt} = (4x-yz)4t \mathbf{i} - (xz)3 \mathbf{j} + (xy)2t \mathbf{k} [/tex]
For the change per meter I will just solve for one of the t eqns, and then substitute back into the equation to find the gradient?
Then for the change per second, it will be the opposite?
 
Last edited:
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  • #2
Nevermind, I figured them out.
 

1. What is a temperature gradient?

A temperature gradient refers to the change in temperature over a distance or space. It measures the rate at which temperature changes, usually in degrees Celsius or Fahrenheit per unit of length.

2. How is a temperature gradient calculated?

To calculate a temperature gradient, you need to measure the difference in temperature between two points and divide it by the distance between those points. This will give you the change in temperature per unit of length, such as degrees Celsius per meter.

3. What factors affect temperature gradient?

The factors that affect temperature gradient include the material or substance, the distance over which the temperature changes, and the rate at which heat is transferred. Other factors may include pressure, altitude, and atmospheric conditions.

4. Why is temperature gradient important?

Temperature gradient is important in many scientific fields, such as physics, geology, and meteorology. It helps us understand how heat is transferred and how temperature changes over distance. It also plays a crucial role in many natural processes and can impact the distribution of plants and animals in an ecosystem.

5. How can temperature gradient be measured?

Temperature gradient can be measured using various instruments, such as thermometers, thermocouples, and infrared cameras. These devices can measure temperature at different points and calculate the change in temperature over a given distance. Additionally, mathematical models and simulations can also be used to estimate temperature gradient in certain situations.

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