The temperature distribution on metal plate is given by

[hargun519]

The temperature distribution on metal plate is given by...

Homework Statement

The temperature distribution on metal plate is given by

T(x,y) = 100/x^2+y^2+1

Calculate the direction derivative in the direction of v= <1,1> at the coordinates (3,2) and at coordinate (3,2) in what direction does the increase, and then decrease most rapidly? Give a unit vector

Homework Equations

The Attempt at a Solution

For the first part of the problem the question asks, where is the plate hottest? I am guessing the plate would be hottest in the center. So, my general idea of this problem is that if we have a disc and then a coordinate plane on the disc, the point 3,2 is on the quadrant 1. The temperature if increasing would point down toward the origin, and then decrease point up, away from the origin.

However, i do not know if I am right, and even if i am i still really don't understand what the problem wants me to do.

 

[Dick]

They want you to find and use the gradient vector of T(x,y). The directional derivative in a direction v at a point (x,y) is given by (v/|v|).grad(T)(x,y) (dot product) for v. If you think about what the dot product means you should be able to figure out that the directions that maximize and minimize that are directions parallel and antiparallel to grad(T)(x,y).

 

[Mark44]

I'm not sure that what you wrote for the temperature is what you meant. I suspect that the function is T(x, y) = 100/(x² + y² + 1), which is different from what you wrote. If that's the case, the highest temperature is NOT in the middle of the plate.

The first part of the problem you posted asks you to find the directional derivative of T in the direction of <1, 1>, evaluated at the point (3, 2).

The second part of the problem you posted asks you to find the directional derivative of T in an arbitrary direction, evaluated at the point (3, 2). From this you are supposed to determine the direction in which the directional derivative is largest and smallest.

So far, your work seems to be entirely based on guesswork, with no apparent evidence that you have tried to calculate anything. Show us what work (not guesses) you have done, and we'll give you a hand.