# [SOLVED]Gradient and scalar function question

#### dwsmith

##### Well-known member
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.