Verify Symmetry: Find $m_{st}$ & $m_{ts}$ in Chain Rule Calculation

[mathmari]

Hey!

Let $f(x, y, z)=e^{xz}\tan (yz)$ and $x=g(s, t)$, $y=h(s, t)$, $z=k(s, t)$. We set $m(s, t)=f(g(s, t), h(s, t), k(s, t))$. Find a formula for $m_{st}$ using the chain rule and verify that the result is symmetric as for $s$ and $t$.

I have done the following:

After using the chain rule I got $$m_{st}=f_{xx}g_{t}g_{s}+f_{xy}(h_{t}g_{s}+g_{t}h_{s})+f_{xz}(k_{t}g_{s}+g_{t}k_{s})+f_{yy}h_{t}h_{s}+f_{yz}(k_{t}h_{s}+h_{t}k_{s})+f_{zz}k_{t}k_{s}f_{x}g_{st}+f_{y}h_{st}+f_{z}k_{st}$$

Is this correct??

How can we verify that the result is symmetric as for $s$ and $t$?? Do we have to find $m_{ts}$ or is there also an other way??

 

[jvicens]

Hi there! Your formula for $m_{st}$ looks correct. To verify that the result is symmetric for $s$ and $t$, we can rewrite the formula in terms of $m_{ts}$ by swapping the subscripts for $s$ and $t$ in each term. If the resulting formula is the same as the one you found for $m_{st}$, then we can say that the result is indeed symmetric. In this case, it would look like:

$$m_{ts}=f_{xx}g_{s}g_{t}+f_{xy}(h_{s}g_{t}+g_{s}h_{t})+f_{xz}(k_{s}g_{t}+g_{s}k_{t})+f_{yy}h_{s}h_{t}+f_{yz}(k_{s}h_{t}+h_{s}k_{t})+f_{zz}k_{s}k_{t}+f_{x}g_{ts}+f_{y}h_{ts}+f_{z}k_{ts}$$

We can see that the formula is indeed the same, just with the subscripts for $s$ and $t$ swapped. This means that the result is symmetric for $s$ and $t$.