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ChiefKeeper92
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ChiefKeeper92 said: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 [itex]\vec{u}[/itex] be a unit vector pointing in the direction of the vector -3,0,1. Use the Chain Rule to show that D[itex]\vec{u}[/itex]f(1,2,3) = 0
c. Explain in geometric terms why D[itex]\vec{u}[/itex]f(1,2,3) = 0.
Homework Equations
D[itex]\vec{u}[/itex]f = ∇f [itex]\bullet[/itex] 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)
ChiefKeeper92 said:#Error
A directional derivative is a measure of how a function changes in a specific direction at a given point. It is defined as the rate of change of the function in the direction of a unit vector. In other words, it calculates the slope of the tangent line in a specific direction at the given point.
In multivariable calculus, differentiability is defined as the ability of a function to be approximated by a linear function at a given point. It means that the function has a well-defined tangent plane at that point, which can be represented by a linear function.
The chain rule in multivariable calculus is a generalization of the single-variable chain rule. It is used to calculate the derivative of a composite function, where the input of one function is the output of another function. The chain rule helps to break down a complex function into simpler components and calculate the derivatives of each component separately.
Partial derivatives are derivatives of a multivariable function with respect to one of its variables, while directional derivatives are derivatives in a specific direction. In other words, partial derivatives measure the rate of change of the function in one direction, while directional derivatives measure the rate of change in a specific direction at a given point.
The chain rule is used to calculate directional derivatives by breaking down a function into simpler components and calculating the derivatives of each component separately. These derivatives are then multiplied by the corresponding partial derivatives of the component functions to get the directional derivative in the desired direction.