Welcome to our community

Be a part of something great, join today!

[SOLVED] Gradient and scalar function question

dwsmith

Well-known member
Feb 1, 2012
1,673
I am trying to determine my scalar function \(f(u_1, u_2, u_3)\) of elliptical cylindrical coordinates.

\begin{align*}
x &= a\cosh(u)\cos(v)\\
y &= a\sinh(u)\sin(v)\\
z &= z
\end{align*}

I have determined my vectors \(\mathbf{U}_u\), \(\mathbf{U}_u\), and \(\mathbf{U}_z\).

\begin{align*}
\mathbf{U}_u &= a\sinh(u)\cos(v)\hat{\mathbf{i}} + a\cosh(u)\sin(v)\hat{\mathbf{j}}\\
\mathbf{U}_v &= -a\cosh(u)\sin(v)\hat{\mathbf{i}} + a\sinh(u)\cos(v)\hat{\mathbf{j}}\\
\mathbf{U}_z &= \hat{\mathbf{k}}
\end{align*}

\[
\nabla f = \frac{1}{h_1}\frac{\partial f}{\partial u_1}\hat{\mathbf{u}}_1 +
\frac{1}{h_2}\frac{\partial f}{\partial u_2}\hat{\mathbf{u}}_2 +
\frac{1}{h_3}\frac{\partial f}{\partial u_3}\hat{\mathbf{u}}_3
\]

I have found \(h_i\)'s as
\[
h_1 = h_2 = \frac{1}{a\sqrt{\cosh^2(u) - \cos^2(v)}}
\]
and
\[
h_3 = 1.
\]

So I need to find \(\frac{\partial f}{\partial u_i}\) but I don't know what my scalar funtion is.