- #1
Solidmozza
- 29
- 1
Hi there,
The Law of Cosines can be stated as
[itex]a^2 = b^2 + c^2 - 2bccos(A)[/itex]
where [itex]a[/itex],[itex]b[/itex], and [itex]c[/itex] are the sides of a triangle, and [itex]A[/itex] is the angle opposite the side [itex]a[/itex]. I have a function, [itex]f(b,c,A)[/itex], with an associated set of partial derivatives [itex](\frac{∂f}{∂c})_{b,A}[/itex] etc. What I want to do is to use a coordinate transformation to get the related derivatives [itex](\frac{∂f}{∂c})_{b,a}[/itex] etc. This looks like a multivariable partial derivative problem with a constraint. Using the chain rule, it seems to me that
[itex](\frac{∂f}{∂c})_{b,A}=\frac{∂f}{∂c}+\frac{∂f}{∂a} \frac{∂a}{∂c}[/itex]
[itex](\frac{∂f}{∂c})_{b,a}=\frac{∂f}{∂c}+\frac{∂f}{∂A} \frac{∂A}{∂c}[/itex]
where I think [itex]\frac{∂f}{∂c}[/itex] is the same in both expressions. I can calculate the [itex]\frac{∂a}{∂c}[/itex] and [itex]\frac{∂A}{∂c}[/itex] parts using the Law of Cosines, but then I don't know what to do with [itex]\frac{∂f}{∂a}[/itex] and [itex]\frac{∂f}{∂A}[/itex], i.e. if I use the transformation [itex]\frac{∂f}{∂a}=\frac{∂f}{∂A}\frac{∂A}{∂a}[/itex] then it looks like [itex](\frac{∂f}{∂c})_{b,A}=(\frac{∂f}{∂c})_{b,a}[/itex] but numerical results and intuition tell me otherwise.
In essence, I don't know how to find the partial derivative [itex](\frac{∂f}{∂c})_{b,a}[/itex] given [itex](\frac{∂f}{∂c})_{b,A}[/itex] and that the variables are connected via the Law of Cosines.
Any assistance with this problem would be greatly appreciated :)
The Law of Cosines can be stated as
[itex]a^2 = b^2 + c^2 - 2bccos(A)[/itex]
where [itex]a[/itex],[itex]b[/itex], and [itex]c[/itex] are the sides of a triangle, and [itex]A[/itex] is the angle opposite the side [itex]a[/itex]. I have a function, [itex]f(b,c,A)[/itex], with an associated set of partial derivatives [itex](\frac{∂f}{∂c})_{b,A}[/itex] etc. What I want to do is to use a coordinate transformation to get the related derivatives [itex](\frac{∂f}{∂c})_{b,a}[/itex] etc. This looks like a multivariable partial derivative problem with a constraint. Using the chain rule, it seems to me that
[itex](\frac{∂f}{∂c})_{b,A}=\frac{∂f}{∂c}+\frac{∂f}{∂a} \frac{∂a}{∂c}[/itex]
[itex](\frac{∂f}{∂c})_{b,a}=\frac{∂f}{∂c}+\frac{∂f}{∂A} \frac{∂A}{∂c}[/itex]
where I think [itex]\frac{∂f}{∂c}[/itex] is the same in both expressions. I can calculate the [itex]\frac{∂a}{∂c}[/itex] and [itex]\frac{∂A}{∂c}[/itex] parts using the Law of Cosines, but then I don't know what to do with [itex]\frac{∂f}{∂a}[/itex] and [itex]\frac{∂f}{∂A}[/itex], i.e. if I use the transformation [itex]\frac{∂f}{∂a}=\frac{∂f}{∂A}\frac{∂A}{∂a}[/itex] then it looks like [itex](\frac{∂f}{∂c})_{b,A}=(\frac{∂f}{∂c})_{b,a}[/itex] but numerical results and intuition tell me otherwise.
In essence, I don't know how to find the partial derivative [itex](\frac{∂f}{∂c})_{b,a}[/itex] given [itex](\frac{∂f}{∂c})_{b,A}[/itex] and that the variables are connected via the Law of Cosines.
Any assistance with this problem would be greatly appreciated :)