# Fixed Variables in Partial Derivatives

• I
• transmini
In summary, the conversation discusses the concept of holding a variable "fixed" when taking partial derivatives and how it affects the outcome. Using the example of ##w=xy## and ##x=yz##, it is shown that the partial derivatives do not match up when different variables are held fixed. The concept is further explained by discussing the parametrisation of a surface and how varying a variable in different parametrisations can result in different displacements. Keeping the variables fixed specifies the parametrisation being used and determines the appropriate variations to be considered.
transmini
My physics book is showing an example of why it matters "what variable you hold fixed" when taking the partial derivative. So it asks to show that
##(\frac{\partial{w}}{\partial{x}})_{y} \neq (\frac{\partial{w}}{\partial{x}})_z##
where ##w=xy## and ##x=yz## and the subscripts are what variable is held fixed.

I'm just not sure what it means by holding a variable "fixed" since all other variables except the two in question are treated as constants, so why would it matter whether ##w## is a function of ##(x, y)## or a function of ##(x, z)##
I mean I see that the partial derivatives don't match up, but don't really see why the variables make a difference since they are the same function.

The condition ##x=yz## describes a surface in three dimensions. Now, you can use different sets of variables to parametrise this surface. One choice is to use x and y and another to use x and z. Varying x in the first parametrisation is generally going to give you a displacement in the surface that will be different from the one you obtain if you vary x in the same way in the second parametrisation. The "keeping these variables fixed" essentially tells you what parametrisation is used and therefore exactly what variations should be considered.

## 1. What are fixed variables in partial derivatives?

Fixed variables in partial derivatives refer to the variables that are held constant during the process of taking a partial derivative. These variables are not affected by the changes in the other variables and remain fixed throughout the calculation.

## 2. Why are fixed variables important in partial derivatives?

Fixed variables are important because they help us isolate the effect of a single variable on a function. By keeping certain variables fixed, we can better understand the relationship between the remaining variables and the function.

## 3. How do you identify fixed variables in partial derivatives?

To identify fixed variables in partial derivatives, we look for the letters or symbols that are not being differentiated. These variables are the ones that remain fixed or constant throughout the calculation.

## 4. Can fixed variables change in partial derivatives?

No, fixed variables cannot change in partial derivatives. They are held constant throughout the process of taking the derivative and their values do not change.

## 5. What is the purpose of keeping certain variables fixed in partial derivatives?

The purpose of keeping certain variables fixed in partial derivatives is to better understand the impact of a single variable on a function. It allows us to analyze how the function changes with respect to that specific variable while keeping all other variables constant.

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