How Can the Potential of a Given Vector Field Be Determined?

[drynada]

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?

 

[mighty2000]

Hello,

To calculate the arc length, you will need to set up an integral with the limits corresponding to the parameter t. For this particular curve, you can use the limits from t=0 to t=1, as these values correspond to the coordinates given in the problem statement. As for verifying that the curve lies on the sphere, you can substitute the expressions for x, y, and z into the equation for a sphere (x^2 + y^2 + z^2 = r^2) and see if it holds true for all values of t.

To determine the potential for the given vector field, you can use the method of separation of variables. First, you can write the given vector field as a gradient of a scalar function, where the scalar function is the potential. This means that the partial derivatives of the potential with respect to the coordinates should match the components of the given vector field. In this case, you will have two equations: f_alpha = partial derivative of the potential with respect to alpha, and f_beta = partial derivative of the potential with respect to beta. You can solve these equations for the potential, and then use the potential to calculate the integral for the conservative vector field.

I hope this helps. Let me know if you have any further questions.