Multivariable partial derivative

In summary: The subtleties are probably there because ##\arctan(x/y)## is multivalued right?In summary, the task was to use the transformation from polar to Cartesian coordinates to show that the partial derivative with respect to x can be expressed as the cosine of the angle φ multiplying the partial derivative with respect to r minus the sine of the angle φ divided by r multiplying the partial derivative with respect to φ. The transformation equations used were x = rcosφ and r = √(x²+y²), and it was necessary to take into account the subtleties of the arctan function in order to solve the problem.
  • #1
RichardJ
4
0

Homework Statement


From the transformation from polar to Cartesian coordinates, show that

\begin{equation}
\frac{\partial}{\partial x} = \cosφ \frac{\partial}{\partial r} - \frac{\sinφ}{r} \frac{\partial}{\partialφ}
\end{equation}

Homework Equations


The transformation from polar to Cartesian coordinates is assumed to be x = r\cosφ

The Attempt at a Solution


To solve the problem i tried to use the multivariable chain rule. Resulting in the following equation:

\begin{equation}
\frac{\partial}{\partial x} =\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partialφ}{\partial x}\frac{\partial}{\partial φ}
\end{equation}

Writing ##r = x/\cosφ## and ##\arccos(x/r) = φ## i tried to solve this problem. But this does not give the right answer.

Am i using the right approach? I think it is necessary to use the multivariable chain rule in some form. But the partial derivative not acting on some other function seems a bit weird to me so i am not sure how to solve this problem.
 
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  • #2
RichardJ said:

Homework Statement


From the transformation from polar to Cartesian coordinates, show that

\begin{equation}
\frac{\partial}{\partial x} = cosφ \frac{\partial}{\partial r} - \frac{sinφ}{r} \frac{\partial}{\partialφ}
\end{equation}

Homework Equations


The transformation from polar to Cartesian coordinates is assumed to be x = rcosφ

The Attempt at a Solution


To solve the problem i tried to use the multivariable chain rule. Resulting in the following equation:

\begin{equation}
\frac{\partial}{\partial x} =\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partialφ}{\partial x}\frac{\partial}{\partial φ}
\end{equation}

Writing ##r = x/cosφ## and ##arccos(x/r) = φ## i tried to solve this problem. But this does not give the right answer.

Am i using the right approach? I think it is necessary to use the multivariable chain rule in some form. But the partial derivative not acting on some other function seems a bit weird to me so i am not sure how to solve this problem.

In LaTeX, standard functions look a lot better if they are preceded by '\', so you get ##\sin \phi## instead of ##sin \phi##, etc.
 
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Likes RichardJ
  • #3
RichardJ said:
##\frac{\partial}{\partial x} =\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partialφ}{\partial x}\frac{\partial}{\partial φ}##
With that equation in mind:
##r=\sqrt{x²+y²}##
##φ=\arctan(\frac{y}{x})## (with some subtleties).
 
  • #4
Ahh, thanks a lot. That solved the problem.
 

1. What is a multivariable partial derivative?

A multivariable partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables, while holding all other variables constant. It is used to analyze how a function changes in response to small changes in its input variables.

2. How do you calculate a multivariable partial derivative?

To calculate a multivariable partial derivative, you take the derivative of the function with respect to the variable of interest, treating all other variables as constants. This is known as a partial derivative because it only considers the change in one variable, not all of them.

3. Why are multivariable partial derivatives important?

Multivariable partial derivatives are important because they allow us to analyze complex functions with multiple variables. They are particularly useful in fields such as physics, economics, and engineering, where variables are often interdependent and changing one can affect the others.

4. What is the difference between a partial derivative and a total derivative?

A partial derivative only considers the change in one variable while holding others constant, while a total derivative considers the change in all variables. This means that a total derivative takes into account the impact of changes in all variables on the function, while a partial derivative does not.

5. How are multivariable partial derivatives used in real-world applications?

Multivariable partial derivatives are used in various real-world applications, such as optimizing production processes in manufacturing, calculating the impact of different factors on economic models, and understanding the behavior of complex systems in physics. They are also used in fields such as machine learning and data analysis to analyze relationships between multiple variables.

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