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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.

Solution Summary

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

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