Product Rule with Partial Derivatives

In summary, the conversation is about solving Laplace's equation using separation of variables and the confusion surrounding the product rule. One person is having trouble understanding the derivatives of f and g, and another person suggests using the product rule to solve the differential equation for r. However, another person argues that it is better to keep the derivative as is to easily solve the equation. After discussing different forms of Bessel's differential equation, they come to the understanding that the notation was causing confusion.
  • #1
Vapor88
24
0
Hi, so I'm trying to solve Laplace's equation by separation of variables, and there's a basic step I'm not understanding with regards to the product rule.

Given
laplace1.png


A product rule (i think) is taken to make the first term easier to deal with and we get

laplace2.png


I'm just having trouble understanding what the derivatives of f and g are. And am I correct in saying that f = 1/r*partial wrt r
g = r * partial phi wrt r

Thank you
 
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  • #2
Vapor88 said:
I'm just having trouble understanding what the derivatives of f and g are. And am I correct in saying that f = 1/r*partial wrt r
g = r * partial phi wrt r

No, to apply the product rule to [itex]\frac{\partial}{\partial r}\left(r\frac{\partial \Phi}{\partial r}\right)[/itex], you let [itex]f(r)=r[/itex] and [itex]g(r)=\frac{\partial \Phi}{\partial r}[/itex].
 
  • #3
If that's the case, then I have a problem. How do you multiply partial derivatives together? I don't see at all how you go from equation one to equation two.
 
  • #4
Hey Vapor,

If you are trying to use separation of variables then your first step should be:
[tex]\phi = f(r)g(\theta)h(z)[/tex].

Coto
 
  • #5
Coto,

I'm okay on that end of things, and understand every step when I plug those equations in, but I just can't get a grasp of this first one.
 
  • #6
Hmm :). Generally speaking, it doesn't make sense to separate that derivative. It makes more sense to keep it as is since you can then easily solve the differential equation that results for r.

In other words, applying the product rule actually moves you back a step!

Anyway, the product rule is:
[tex]\frac{d}{dr}(f(r) \cdot g(r)) = g(r)\frac{df}{dr} + f(r)\frac{dg}{dr}[/tex].

Use this in combination with gabbas advice, where,
[tex]\frac{d}{dr}\frac{d\phi}{dr} = \frac{d^2\phi}{dr^2}[/tex].
 
  • #7
As coto says, the product rule tells you that

[tex]\frac{\partial}{\partial r}\left(r\frac{\partial \Phi}{\partial r}\right)=\frac{\partial}{\partial r}\left(f(r)g(r)\right)=\frac{\partial f}{\partial r}g(r)+f\frac{\partial g}{\partial r}[/tex]

So, calculate [tex]\frac{\partial f}{\partial r}[/itex] and [tex]\frac{\partial g}{\partial r}[/itex] and see what that gives you.
 
  • #8
Coto said:
Hmm :). Generally speaking, it doesn't make sense to separate that derivative. It makes more sense to keep it as is since you can then easily solve the differential equation that results for r.

That depends on what form you are used to seeing Bessel's differential equation in.
 
  • #9
gabbagabbahey said:
That depends on what form you are used to seeing Bessel's differential equation in.

I suppose. I strongly encourage the "Sturm-Liouville" form.
 
  • #10
Excellent. I now understand. I guess I was just having a problem with the notation of it. I didn't understand that the partial wrt r was for what was in the parentheses after it, and that the 1/r term is simply multiplied through afterwards. Thanks!
 

What is the product rule with partial derivatives?

The product rule with partial derivatives is a mathematical rule used to find the derivative of a product of two or more functions that each depend on multiple variables. It is commonly used in multivariate calculus to solve problems involving rates of change.

How do you apply the product rule with partial derivatives?

To apply the product rule with partial derivatives, you first take the derivative of each individual function with respect to each variable. Then, you multiply each of these derivatives by the other function and add them together. This gives you the derivative of the entire product function.

What is the formula for the product rule with partial derivatives?

The formula for the product rule with partial derivatives is d(uv)/dx = u*(dv/dx) + v*(du/dx). This can also be written as d(uv)/dy = u*(dv/dy) + v*(du/dy) for functions with multiple variables.

When is it necessary to use the product rule with partial derivatives?

The product rule with partial derivatives is necessary when finding the derivative of a product function that involves multiple variables. It is also used when taking the derivative of a function that is a combination of two or more simpler functions.

What are some common applications of the product rule with partial derivatives?

The product rule with partial derivatives is commonly used in physics, engineering, and economics to solve problems involving rates of change. It is also used in optimization problems, such as finding the maximum or minimum value of a function with multiple variables.

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