How Do You Solve for At Using Subscript Derivative Notation?

[Abdul.119]

I have an equation that looks like

A_{t, r} = A_{φ, r}

If I know that A_φ = r⁴ , then how do I find A_t ?

I believe that the above equation is equivalent to: ∂/∂r (A_t) = ∂/∂r (A_φ) , correct?

Then substitute the value A_φ and we have ∂/∂r (A_t) = ∂/∂r (r⁴)

And then to get A_t I take the integral on both sides? so eventually A_t = r⁵/5 ?

 

[Mark44]

I have an equation that looks like

A_{t, r} = A_{φ, r}

If I know that A_φ = r⁴ , then how do I find A_t ?

I believe that the above equation is equivalent to: ∂/∂r (A_t) = ∂/∂r (A_φ) , correct?

Yes, $A_{t, r}$ is notation that means $\frac{\partial}{\partial r}\frac{\partial A}{\partial t}$

Then substitute the value A_φ and we have ∂/∂r (A_t) = ∂/∂r (r⁴)

And then to get A_t I take the integral on both sides?

Yes. Integrate with respect to r.

so eventually A_t = r⁵/5 ?

Are r and t independent of each other. If so, the above looks fine.