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Solutions to Various Problems in Multivariable Calculus

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Write out the chain rule for each of the following functions and justify your answer in each case using this following Theorem.
Theorem: Chain Rule
Let U ⊂ R^m⟶R^p and f:V⊂ R^m⟶R^p be given functions such that g maps U into V, so that f∘g is defined. Suppose g is differentiable at X_0 and f is differentiable at y_0=g(X_0). Then f∘g is differentiable at X_0 and D(f∘g)( X_0 )=Df(y_0 )Dg(X_0). The right-hand side is the matrix product of Df(y_0 ) with Dg(X_0).
δh/δx where h(x,y)=f(x,u(x,y))
dh/dx where h(x)=f(x,u(x),v(x))
δh/δx where h(x,y,z)=f(u(x,y,z),v(x,y),w(x))
Verify the chain rule for δh/δx, where h(x,y)=f(u(x,y),v(x,y)) and
f(u,v)=(u^2+v^2)/(u^2-v^2 ) , u(x,y)=e^(-x-y) , v(x,y)=e^xy.
Suppose that the temperature at the point (x,y,z) in space is T(x,y,z)=x^2+y^2+z^2. Let a particle follow the right-circular helix σ(t)=(cos⁡〖t,sin⁡〖t,t)〗 〗 and let T(t) be its temperature at time t. What is T^' (t)?
Captain Ralph is in trouble near the sunny side of Mercury. The temperature of the ship's hull when he is at location (x,y,z) will be given by T(x,y,z)=e^(-x^2-2y^2-3z^2 ), where x,y, and z are measured in meters. He is currently at (1,1,1).
In what direction should he proceed in order to decrease the temperature most rapidly?
If the ship travels at e^8 meters per second, how fast will be the temperature decrease if the proceeds in that direction?
Unfortunately, the metal of the hull will crack if cooled at a rate greater than √14 e^2 degrees per second. Describe the set of possible directions in which he may proceed to bring the temperature down at no more than that rate.

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Solution Summary

We solve various problems in multivariable calculus, including the chain rule and problems which work with gradients.

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