Line Integral dl in spherical polar coordinates

[wam_mi]

Homework Statement

Hi guys,

I'm trying to evaluate a line integral, Integration of Vector A dot dL

The vector A was given to be a function of r, theta and fi in spherical polar coordinates.

The question states that an arbitrary closed loop C is the circle parametrised by fi at some arbitrary values of (r, theta).

I would like to ask the following questions:

(i) How do I rewrite dl, the line element, in terms of spherical polar co-ordinates? What does it mean by the closed curve C is the circle parametrised by fi at some arbitrary values of (r, theta). How does it change my dl if C wasn't a circle?

(ii) What are the limits of the line integral?



Thanks guys!

Homework Equations

The Attempt at a Solution

 

[lippyka]

(i) dl in terms of spherical polar coordinates can be written as:dl = (r sin(θ) cos(φ))dφ + (r sin(θ) sin(φ))dθ + (r cos(θ))drThe closed curve C being parametrised by fi at some arbitrary values of (r, theta) means that the line integral is taken over a circle in spherical polar coordinates, with a radius of r, an elevation of θ and azimuth of φ. If C wasn't a circle, the form of dl would still be the same, but the limits of the line integral would change accordingly.(ii) The limits of the line integral depend on the closed curve C chosen. Since C is a circle, the limits are given by the start and end points of the path. In this case, the limits of the line integral would be from 0 to 2π.