# Potential of the vector field

## Homework Statement

I have a curve $$\Psi(t) = \hat h_\alpha$$ where the coordinates are $$\alpha=0, \beta=t$$ and $$\gamma=t$$ in the system. Additionaly

$$x=\sqrt2 ^\alpha \cdot(sin\beta-cos\beta)\cdot \frac{1}{cosh\gamma}$$

$$y=\sqrt2 ^\alpha \cdot(cos\beta+sin\beta)\cdot \frac{1}{cosh\gamma}$$

$$z=\sqrt2 ^{\alpha+1} \cdot tanh\gamma$$

## Homework Equations

$$x^2 + y^2 +z^2 =r^2$$

## The Attempt at a Solution

My job was to derived all normalized frame vectors of this system I did it. Later calculate the arc length $$s$$ and verify that the curve is lying on the sphere.

My problem is:
1) How can I define the limits for the integral to calculate the arc length? Is the condition for the curve which is lying on the sphere correct?

$$x^2 + y^2 +z^2 =r^2$$

2) Last issue.
I need to assume the vector field

$$G(\alpha,\beta,\gamma)={cosh\gamma\over \sqrt{2}^{\alpha+1}cos\beta} \hat h_\beta+{sinh\gamma\over \sqrt{2}^{\alpha+1}cos\beta} \hat h_\gamma$$

to be a conservative and determine the corresponding potential $$\phi$$.

How can I determine this potential.
Usually I always had 3 factors $$f_x, f_y, f_z$$ here $$f_\alpha, f_\beta, f_\gamma$$ where I was calculating integral of $$f_x$$. Later I was calculating derivatives for $$y$$ and $$z$$ from result of integral of $$f_x$$. I don't know how should I manage it with only two factors. In the beginning I just assumed that $$\int h_\alpha = 0 + C_1(\beta,\gamma)$$ but in the end I'm not getting correct result which should be $$\Phi=ln|cosh(\gamma) \cdot tan(\frac{\Pi}{4}+ \frac{\beta}{2})|+const.$$

What should I do?