Multivariable Calculus: Applications of Grad (and the Chain Rule?)

In summary: Then, \mathbf{x} \cdot \nabla f(\mathbf{x}) = p f(\mathbf{x}) as required.In summary, a differentiable function f: ℝn → ℝ is homogenous of degree p if for every x ∈ ℝn and every a > 0, f(ax) = apf(x). If f is homogenous, then x · ∇f(x) = pf(x). This can be proven by differentiating f(ax) with respect to a and letting a = 1, which leads to x · ∇f(x) = pf(x).
  • #1
gadje
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0

Homework Statement


We say that a differentiable function [tex]f : \mathbb{R}^n \rightarrow \mathbb{R}[/tex] is homogenous of degree p if, for every [tex]\mathbf{x} \in \mathbb{R}^n [/tex] and every a>0,
[tex]f(a\mathbf{x}) = a^pf(\mathbf{x}).[/tex]

Show that, if f is homogenous, then [tex]\mathbf{x} \cdot \nabla f(\mathbf{x}) = p f(\mathbf{x})[/tex] .

Homework Equations


The chain rule (not sure if I need it): [tex]\displaystyle \frac{d}{dt} f(\mathbf{x}(t)) = \Sigma_{i = 1}^{n} f_{x_i}\dot{x_i} = \dot{\mathbf{x}} \cdot \nabla f[/tex]

The Attempt at a Solution



Well, I see the resemblance between the rightmost hand side of the chain rule I wrote down, but I don't really understand how the chain rule is applied in this situation, seeing as there isn't anything about x being a function of something else here.

Any ideas?
Cheers.
 
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  • #2
Try differentiating

[tex]f(a\mathbf{x}) = a^pf(\mathbf{x})[/tex]

with respect to a.
 
  • #3
[tex]\frac{\partial}{\partial a} f(a \mathbf{x}) = pa^{p-1}f(\mathbf{x})[/tex]

Okay. I'm still clueless.

EDIT:

Hang on. [tex] \frac{\partial}{\partial a} f(a \mathbf{x}) = \frac{\partial}{\partial a} (a \mathbf{x}) \frac{\partial}{\partial \mathbf{x}} f(a\mathbf{x}) = \mathbf{x} \cdot \nabla f (a \mathbf{x}) [/tex] (if you'll forgive the abuse of notation), right?
 
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  • #4
Good. Now let a=1.
 

FAQ: Multivariable Calculus: Applications of Grad (and the Chain Rule?)

What is multivariable calculus?

Multivariable calculus is a branch of mathematics that deals with the calculus of functions with multiple variables. It extends the concepts of single variable calculus to functions with more than one independent variable.

What are the applications of multivariable calculus?

Multivariable calculus has many applications in fields such as physics, engineering, economics, and computer graphics. It is used to model and analyze complex systems that involve multiple variables.

What is the grad function in multivariable calculus?

The grad function, also known as the gradient, is a vector that represents the direction and magnitude of the steepest slope of a multivariable function at a given point. It is used to find the direction of maximum change of a function.

How is the chain rule used in multivariable calculus?

The chain rule is a fundamental rule in multivariable calculus that allows us to differentiate composite functions. In other words, it helps us find the derivative of a function that is composed of multiple functions.

What are some real-world examples of using multivariable calculus?

Multivariable calculus is used in various real-world applications, such as calculating trajectories of objects in motion, optimizing production processes in economics, and creating 3D computer graphics. It is also used in physics to analyze the behavior of multiple interacting particles.

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