Multivariable Calculus: Directional Derivatives, Differentiablity, Chain Rule

[ChiefKeeper92]

#Error

 

[Ray Vickson]

Homework Statement

Suppose g : ℝ→ℝ is a twice differentiable function. Define f :R³ → ℝ by
f(x,y,z)=g(x^2 +y^2 +z^2).

a. Show that f is differentiable using an analog to the theorem : If the partial derivatives of x and y exist near (a,b) and are continuous at (a,b) then f is differentiable at (a,b).

b. Let $\vec{u}$ be a unit vector pointing in the direction of the vector -3,0,1. Use the Chain Rule to show that D$\vec{u}$f(1,2,3) = 0

c. Explain in geometric terms why D$\vec{u}$f(1,2,3) = 0.

Homework Equations

D$\vec{u}$f = ∇f $\bullet$ u = abs(∇f) abs(u) cosθ = abs(∇f) cosθ.

∇f=∂f/∂x i + ∂f/∂y j + ∂f/∂z k

dz/dt = (∂f/∂x)*(dx/dt) + (∂f/∂y)*(dy/dt)

Just use the equations you have written down, or explain why you are unable to use them.

RGV

 

[berkeman]

#Error

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