What does the line integral of a vector field represent?
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 a line integral tell us?
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.
How do you interpret the curl of a vector field?
The curl of a vector field is a vector field. The curl of a vector field at point P measures the tendency of particles at P to rotate about the axis that points in the direction of the curl at P. A vector field with a simply connected domain is conservative if and only if its curl is zero.
What is the divergence of a vector field how can you interpret it?
The divergence of a vector field simply measures how much the flow is expanding at a given point. It does not indicate in which direction the expansion is occuring. Hence (in contrast to the curl of a vector field), the divergence is a scalar.
Are line integrals path dependent?
If the value of the line integral changes from one curve to the next, then the vector field is path dependent.
How we know the line integral is independent of path?
Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.
What does divergence and curl of a vector signify?
The divergence and curl of a vector field are two vector operators whose basic properties can be understood geometrically by viewing a vector field as the flow of a fluid or gas. The curl of a vector field captures the idea of how a fluid may rotate. Imagine that the below vector field F represents fluid flow.
Can we define divergence over all vector valued function?
The divergence is an operator, which takes in the vector-valued function defining this vector field, and outputs a scalar-valued function measuring the change in density of the fluid at each point.
What does the zero value for the divergence of a vector field imply?
It means that if you take a very small volumetric space (assume a sphere for example) around a point where the divergence is zero, then the flux of the vector field into or out of that volume is zero. In other words, none of the arrows of the vector field will be piercing the sphere.
Can line integrals be zero?
You can interpret the line integral being zero to have some special meaning: If we now move the object along a given path and the path integral is zero, then we didn’t need to use any work to do it, i.e. we didn’t need to work against the force field.
What is the value of the line integral?
In a two-dimensional field, the value at each point can be thought of as a height of a surface embedded in three dimensions. The line integral of a curve along this scalar field is equivalent to the area under a curve traced over the surface defined by the field. The length of the line can be determined by the sum of its arclengths
What is irrotational vector field?
An irrotational vector field is a vector field that has a curl of zero. Now curl is related to the actual rotation of the vector field, though you should not accept that curl is zero just because it looks like the vector field doesn’t rotate.
What is an electric field vector?
Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge, by an infinitesimal test charge at that point.