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\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.