What do you mean by line integral?
A line integral (sometimes called a path integral) is the integral of some function along a curve. These vector-valued functions are the ones where the input and output dimensions are the same, and we usually represent them as vector fields.
What does Green’s theorem calculate?
In summary, we can use Green’s Theorem to calculate line integrals of an arbitrary curve by closing it off with a curve C0 and subtracting off the line integral over this added segment. Another application of Green’s Theorem is that is gives us one way to calculate areas of regions.
What does it mean when a line integral is 0?
A line integral in a vector field evaluated over a closed loop will be 0 for all possible curves if and only if (iff) the vector field is conservative.
What is line integral of magnetic field?
The line integral of a magnetic field around a closed path is equal to the total current flowing through the area bounded by the contour (Figure ).
Why line integral is used?
A line integral allows for the calculation of the area of a surface in three dimensions. Or, in classical mechanics, they can be used to calculate the work done on a mass m moving in a gravitational field. Both of these problems can be solved via a generalized vector equation.
What is green and Stokes theorem?
Stokes’ theorem is a generalization of Green’s theorem from circulation in a planar region to circulation along a surface. Green’s theorem applies only to two-dimensional vector fields and to regions in the two-dimensional plane. Stokes’ theorem generalizes Green’s theorem to three dimensions.
How do I apply Greens theorem?
Warning: Green’s theorem only applies to curves that are oriented counterclockwise. If you are integrating clockwise around a curve and wish to apply Green’s theorem, you must flip the sign of your result at some point.
How do you solve a line integral?
Evaluating Line Integrals Fortunately, there is an easier way to find the line integral when the curve is given parametrically or as a vector valued function. We will explain how this is done for curves in R2; the case for R3 is similar. ds=||r′(t)||dt=√(x′(t))2+(y′(t))2.
What is the curl of magnetic field?
curl B·da = J·da Thus the curl of a magnetic field at any point is equal to the current density at that point. This is the simplest statement relating the magnetic field and moving charges.
How do line integrals work?
a) How line integrals arise. The figure on the left shows a force F being applied over a displacement Δr. Work is force times distance, but only the component of the force in the direction of the displacement does any work. So, work = |F| cos θ |Δr| = F · Δr.