What is the difference between integral and line integral?
A line integral is an integral where the function to be integrated is evaluated along a curve and a surface integral is a generalization of multiple integrals to integration over surfaces.
What is a scalar line integral?
A line integral (sometimes called a path integral) of a scalar-valued function can be thought of as a generalization of the one-variable integral of a function over an interval, where the interval can be shaped into a curve. This function describes how the slinky might be thicker in some parts and thinner in others.
Do you think the integral is path independent What is the condition the terms in the integral equation needs to satisfy that the integral is path independent?
An integral is path independent if it only depends on the starting and finishing points. Consequently, on any curve C={r(t)|t∈[a,b]}, by the fundamental theorem of calculus ∫CFdr=∫C∇fdr=f(r(b))−f(r(a)), in other words the integral only depends on r(b) and r(a): it is path independent.
What is the integral of a vector?
From the definition of an integral, this means that a vector-valued function describing the position of an object is the integral of the vector-valued function that describes velocity of the same object.
What is the difference between line integral and double integral?
The line integral involves a vector field and the double integral involves derivatives (either div or curl, we will learn both) of the vector field. First we will give Green’s theorem in work form. The line integral in question is the work done by the vector field. The double integral uses the curl of the vector field.
What is a vector 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.
How that the line integral is independent of path and evaluate the integral?
The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path. If ∫C→F⋅d→r=0 ∫ C F → ⋅ d r → = 0 for every closed path C then ∫C→F⋅d→r ∫ C F → ⋅ d r → is independent of path.
Does line integral depend on path?
Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.
What is the difference between line integrals over scalar fields?
Line integrals over vector fields share the same properties as line integrals over scalar fields, with one important distinction. The orientation of the curve matters with line integrals over vector fields, whereas it did not matter with line integrals over scalar fields. It is relatively easy to see why.
What is a line integral and why is it useful?
A line integral gives us the ability to integrate multivariable functions and vector fields over arbitrary curves in a plane or in space. There are two types of line integrals: scalar line integrals and vector line integrals.
Does the orientation of the curve matter for line integrals?
The orientation of the curve matters with line integrals over vector fields, whereas it did not matter with line integrals over scalar fields. It is relatively easy to see why. Let be the unit circle. The area under a surface over is the same irrespective of orientation.
What is a scalar field?
Just as a vector field is defined by a function that returns a vector, a scalar field is a function that returns a scalar, such as . We waited until now to introduce this terminology so we could contrast the concept with vector fields. We formally define this line integral, then give examples and applications.