(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hello all,

New to partial derivatives. I was wondering if someone could look over my work and determine if my final step is as far as I can take the proble (ie. that will be my solution). Thanks in advance.

Let the temperature of a 2D domain in polar coordinates [itex](r, \varphi)[/itex] be given by [itex]T=f(r, \varphi, t)[/itex], where t is time. Suppose a probe follows the straight path, given in Cartesian coordinates by [itex]x=X(t), y=Y_0=constant[/itex]. Using the fact that [itex]r^2=x^2+y^2, \varphi=arctan(y/x)[/itex], find [itex]dT/dt[/itex].

3. The attempt at a solution

[itex]\frac{dr}{dt}=\frac{\delta r}{\delta x}\frac{dx}{dt}+\frac{\delta r}{\delta y}\frac{dy}{dt}=\frac{\delta r}{\delta x}\frac{dx}{dt}[/itex]

and

[itex]\frac{d\varphi}{dt}=\frac{\delta \varphi}{\delta x}\frac{dx}{dt}+\frac{\delta\varphi}{\delta y}\frac{dy}{dt}=\frac{\delta\varphi}{\delta x}\frac{dx}{dt}[/itex]

thus,

[itex]\frac{dT}{dt}=\frac{\delta T}{\delta r}\frac{dr}{dt}+\frac{\delta T}{\delta\varphi}\frac{d\varphi}{dt}+\frac{\delta T}{\delta t}[/itex]

Next, we have,

[itex]\frac{dT}{dt}=\frac{\delta T}{\delta r}\frac{\delta r}{\delta x}\frac{dx}{dt}+\frac{\delta T}{\delta\varphi}\frac{\delta\varphi}{\delta x}\frac{dx}{dt}+\frac{\delta T}{\delta t}[/itex]

[itex]=\frac{\delta T}{\delta r}(x)\frac{d[X(t)]}{dt}-\frac{\delta T}{\delta\varphi}\frac{1}{1+(y/x)^2}\frac{y}{x^2}\frac{d[X(t)]}{dt}+\frac{\delta T}{\delta t}[/itex]

Is this as far as I can take it with the information given?

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# Partial Differentiation

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