Question

The function T = x^2 + 2y^2 + 2x^2 gives temperature at each point in space. What shape are isotherms? and also, at point (1,1,1), which direction should I go to get the most rapid decrease in T?

Can someone check my working for the second part?

Answer 1

Differentiate wrt to x, fx = 2x
Differentiate wrt to y, fy = 4y
Differentiate wrt to z, fz = 4z
At (1,1,1), gradient vector = 2(i+2j+2k)
u = 1/sqrt(1+2^2+2^2) (i + 2j + 2k)
Hence rapid decrease in direction of -1/3 i - 2/3 j - 2/3k?

I'm not sure what isotherms are, how does one sketch level curves?

Answer 2

Isotherm refers to the level sets(in this case level surfaces). Look at x^2+y^2+2z^2=k, what kind of level curves do you think you will get from such an eqn?

Answer 3

Also the direction of maximum rate of change is in the direction of the gradient vector, so what you have so far is correct.

Answer 4

Ellipsoid was the answer I eventually obtained. Is that right? Cheers

Answer 5

your equation is such like T=x^2+2Y^2+2z^2
for isotherm T=constant let it k, so ,equation will be x^2+2y^2+2z^2=k
that's an elliptoid with origin (0,0,0). now find gradient at (1,1,1) that will be direction of max. change. now taking two point on two opposite side of (1,1,1) on line whose direction will be same as gradient and cheak where temp increase or decrease .the point where temp. will dec . and (1,1,1) you will get a vector whose tail will be on(1,1,1) and that will be direction.. it's done

Answer 6

Thanks Abhi