# Multivariable calculus, partial derivatives

• Feodalherren
In summary, the conversation is about the chain rule and understanding the notation used in a problem. One person is confused and another person helps clarify by using more specific notation and explaining the chain rule. They also discuss how to evaluate the partial derivatives in the problem.
Feodalherren

## The Attempt at a Solution

Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

Are you familiar with the chain rule?

Of course. The notation in this problem is so confusing that I can't follow what's happening.

Let's use the notation ##G(s,t) = (u(s,t), v(s,t))##.

What you need to do is to compute

$$\frac{\partial}{\partial s} F\circ G$$

What will that be according to the chain rule?

Correct?

What happened to ##F##? I can only see ##G## showing up.

Ok I have no idea.. I can't remember what FoG means :/. You didn't even define F as anything?

Feodalherren said:
Ok I have no idea.. I can't remember what FoG means :/. You didn't even define F as anything?

##F## is just an arbitrary differentiable map (as shown in the problem statement).

What ##F\circ G## means is that it is the map which sends ##(s,t)## to ##F(G(s,t))##.

Can you show me the chain rule you've learned (and perhaps the variations)?

1 person
I've only learned the one that I demonstrated. You draw one of those trees and then just differentiate however many times you need to in order to get to the variable that you need.

I'm totally lost on the notation yet again.

Feodalherren said:

## The Attempt at a Solution

Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

Is the "subscript" notation throwing you off? If so, forget it and use a more exact nomenclature: ##W = F(u,v)## gives
$$\frac{\partial W}{\partial s} = \frac{\partial F}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial F}{\partial v} \frac{\partial v}{\partial s}\: \longleftarrow \text{ chain rule}\\ \text{ }\\ \text{or}\\ \text{ }\\ W_s = F_u u_s + F_v v_s$$
etc.

1 person
Ok that helps Ray. Now for the evaluation.

If I know Ws (1,0) then
s=1, t=0

Then I get:
u (1,0) and s (1,0).

Where do I evaluate the first therm dF/du?

## 1. What is multivariable calculus?

Multivariable calculus is a branch of mathematics that deals with the study of functions of multiple variables. It involves the use of derivatives, integrals, and vector calculus to understand and analyze how multiple variables affect a function.

## 2. What are partial derivatives?

Partial derivatives are a type of derivative that calculates the rate of change of a function with respect to one of its variables, while holding the other variables constant. It helps in understanding how a small change in one variable affects the overall function.

## 3. How are partial derivatives used in multivariable calculus?

Partial derivatives are used in multivariable calculus to find the critical points of a function, which are points where the partial derivatives are equal to zero. This helps in determining the maximum and minimum values of a function, as well as the direction in which the function is increasing or decreasing.

## 4. What is the difference between partial derivatives and ordinary derivatives?

The main difference between partial derivatives and ordinary derivatives is that partial derivatives deal with functions of multiple variables, while ordinary derivatives deal with functions of a single variable. Partial derivatives also involve holding other variables constant, while ordinary derivatives do not.

## 5. How is multivariable calculus used in real-world applications?

Multivariable calculus has various real-world applications such as in physics, economics, engineering, and computer science. It is used to model and analyze complex systems involving multiple variables, such as in optimization problems, motion of objects, and financial models.

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