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Gradient Question

  • #1

Homework Statement




The temperature in a room is given by T(x,y,z)=2x^2+3y^2 - 4z. A flying bug located at P(1,1,2) in the room desires to fly in such a direction that it will warm as soon as possible. In what direction must the bug fly?

Taking the gradient of T: 4x(x hat)+6y(y hat)-4(z hat). Do I plug the P(1,1,2) into that equation to determine the direction? Doing this, the answer would be. 1/SQRT(68) * [4(x hat) + 6(y hat)-4(z hat)]

But what if P=(2,2,2). Would if be: 1/SQRT(224) * [8(x hat)+12(y hat) + 4(z hat)]
 

Answers and Replies

  • #2
LowlyPion
Homework Helper
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Homework Statement




The temperature in a room is given by T(x,y,z)=2x^2+3y^2 - 4z. A flying bug located at P(1,1,2) in the room desires to fly in such a direction that it will warm as soon as possible. In what direction must the bug fly?

Taking the gradient of T: 4x(x hat)+6y(y hat)-4(z hat). Do I plug the P(1,1,2) into that equation to determine the direction? Doing this, the answer would be. 1/SQRT(68) * [4(x hat) + 6(y hat)-4(z hat)]

But what if P=(2,2,2). Would if be: 1/SQRT(224) * [8(x hat)+12(y hat) + 4(z hat)]
I'd say basically correct. Looks like a typo in the last term.

∇T of 2x2+ 3y2 - 4z = (4x,6y,-4)
 

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