This discussion focuses on sketching isotherms for the temperature field defined by the equation $T=\cos(x)\sinh(y)$. To create isotherms, one must solve the equation $C=\cos(x)\sinh(y)$ for various constants $C$. The solution for $y$ is given by $y=\text{arcsinh}(C\sec(x))$. By selecting multiple values for $C$ and plotting the resulting function, users can effectively visualize the isotherms in the $(x,y)$ plane.
PREREQUISITESStudents and professionals in mathematics, physics, and engineering who are interested in visualizing temperature distributions and understanding isothermal processes.
Ackbach said:Presumably, you actually have a temperature field, right? That is, $T=\cos(x)\sinh(y)$. In that case, you want to sketch whole bunch of curves in the $(x,y)$ plane of the form $C=\cos(x)\sinh(y)$. Why a constant? Because you're after the isotherms - the prefix "iso" meaning "same". In fact, you can solve this equation for $y$:
\begin{align*}
C&=\cos(x)\sinh(y) \\
\sinh(y)&=C \sec(x) \\
y&=\text{arcsinh}(C\sec(x)).
\end{align*}
So, pick a few $C$'s, plot the above function, and you've got your isotherms.