Flux Integral: How to find ds for line integrals in general

[Marcin H]

Homework Statement

Homework Equations

flux = int(b (dot) ds)

The Attempt at a Solution

I just wanted clarification on finding ds. I understand why ds is in the positive yhat direction (just do rhr) but I don't understand where the dxdz come from. How do we find ds in general?

 

[haruspex]

but I don't understand where the dxdz come from.

It s just the area of a rectangular element, dx by dz, in the xz plane. In polar it would have been rdrdθ, etc.

 

[Marcin H]

It s just the area of a rectangular element, dx by dz, in the xz plane. In polar it would have been rdrdθ, etc.

So what is the general procedure for finding ds? I feel like I struggle determining ds for problems.

 

[haruspex]

So what is the general procedure for finding ds? I feel like I struggle determining ds for problems.

It depends on your coordinate system. With general coordinates, ξ, η say, you consider the product dξdη. In general, the area enclosed by the points (ξ,η), (ξ+dξ,η), (ξ+dξ,η+dη), (ξ,η+dη) might have area equal to dξdη. To make the right area you may need to multiply by a Jacobian.

In the case of Cartesian coordinates dxdy does have the right area.
For plane polar coordinates you need a factor r. This works because the increments dr and rdθ are at right angles, so form a near-enough rectangle.
For the surface of a sphere in spherical polar a different Jacobian is needed.