Simplifying Partial Derivatives in Multivariable Calculus

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In summary, the conversation discusses simplifying two expressions involving partial derivatives and finding a commutator. The first expression simplifies to y*d/dx, while the second one simplifies to zx*d^2/dydz. There is some confusion regarding the placement of parentheses and the exact question, but the correct approach is to treat each expression as a product of derivatives acting on an arbitrary function.
  • #1
hoch449
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Homework Statement



Simplify the following two expressions:

[tex]y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}[/tex]

[tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}[/tex]


The Attempt at a Solution



for the first one: [tex]y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}[/tex]

[tex]\frac{\partial}{\partial z}z = 1[/tex]

so therefore [tex]y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}= y(1)\frac{\partial}{\partial x}= y\frac{\partial}{\partial x}[/tex]

How come this is incorrect?




For the second one: [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}[/tex]

I cannot write that [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}= x\frac{\partial}{\partial z}z\frac{\partial}{\partial y}[/tex]

so therefore [tex]z\frac{\partial}{\partial y}x\frac{\partial}{\partial z}= zx\frac{\partial^2}{\partial y\partial z}[/tex]

The second one I believe is correct though..
 
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  • #2
You need to be a little more clear on your notation. Suppose that f is some arbitrary function that we can let the expressions act on.
Then by
[tex]
\left( y\frac{\partial}{\partial z}z\frac{\partial}{\partial x} \right) f
[/tex]
do you mean
[tex]
y\frac{\partial}{\partial z}\left( z \frac{\partial f}{\partial x} \right)
[/tex]
or, as you imply in your first post,
[tex]
y\left( \frac{\partial}{\partial z}z\right)\frac{\partial f}{\partial x}
[/tex]
or yet something else?
 
  • #3
Depends on where you put the parentheses. If the first one means y*d/dz(z*d/dx) you need to use a product rule on the z*d/dx product. To make it clearer write it as y*d/dz(z*df/dx) where f is a test function.
 
  • #4
I don't believe there are any specific parentheses. The exact question is to find the following commutator [tex][Lx,Ly][/tex]

where:

[tex]Lx= (y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y})[/tex]

[tex]Ly= (z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z})[/tex]
 
  • #5
In that case you should treat it as
[tex]

y\frac{\partial}{\partial z}\left( z \frac{\partial f}{\partial x} \right)

[/tex]
 

1. What does it mean to "simplify an expression?"

Simplifying an expression means to reduce it to its simplest form by combining like terms, removing parentheses, and following the order of operations.

2. Why is it important to simplify expressions?

Simplifying expressions can make them easier to understand and work with, especially when solving equations or simplifying larger expressions. It can also help to identify patterns and relationships between terms.

3. What are the basic rules for simplifying expressions?

The basic rules for simplifying expressions include combining like terms, using the distributive property, and following the order of operations (PEMDAS).

4. How do you simplify expressions with variables?

To simplify expressions with variables, you can follow the same rules as simplifying expressions with numbers, such as combining like terms and using the distributive property. Make sure to also follow any additional rules for simplifying specific types of expressions, such as factoring or using exponent rules.

5. Can expressions always be simplified?

No, not all expressions can be simplified. Some expressions may already be in their simplest form or may not have any like terms to combine. It is important to recognize when an expression cannot be simplified further.

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