How do you find the gradient of a line integral?

How do you find the gradient of a line integral?

Rewriting this expression in terms of the original curve C from point P to point Q, we obtain the gradient theorem for line integrals: ∫C∇f⋅ds=f(Q)−f(P).

What does a line integral tell you?

In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve.

How do you parameterize a vector field?

To compute the work, parameterize the curve C by the vector function r(t)= with a<=t<=b, where r(a) is the initial point and r(b) is the final point. Let us consider the work required to move the object on an infinitesimal piece of the curve from position r(t) to r(t+dt).

Can a line integral be negative?

If the integral is positive, you will arrive in Atlanta early, if the integral is negative you will arrive late. This sort of integral is called a line integral or path integral because we are integrating along a line or a path.

What does it mean when a line integral is 0?

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.

Can a line integral be zero?

What is line integral used for?

A line integral is used to calculate the surface area in the three-dimensional planes. Some of the applications of line integrals in the vector calculus are as follows: A line integral is used to calculate the mass of wire. It helps to calculate the moment of inertia and centre of mass of wire.

What is the use of line integral?

A line integral is used to calculate the mass of wire. It helps to calculate the moment of inertia and centre of mass of wire. It is used in Ampere’s Law to compute the magnetic field around a conductor. In Faraday’s Law of Magnetic Induction, a line integral helps to determine the voltage generated in a loop.

What is a line integral used for?

A line integral allows for the calculation of the area of a surface in three dimensions. Line integrals have a variety of applications. For example, in electromagnetics, they can be used to calculate the work done on a charged particle traveling along some curve in a force field represented by a vector field.

Is line integral positive or negative?

is positive. It can be shown that the value of the line integral is independent of the speed that the curve is drawn by the parameterization. is negative, because the tangent vectors of the path are going “against” the field vectors.

When to use variable name in line integral?

It is common to use the variable name to refer to the component functions when you parametrize a curve, so x = x (t) and y = y (t). Here x (t) = g (t) and y (t) = h (t) defined earlier in the video.

Is there an introduction to the line integral?

However, not all listeners, in fact most I dare say, are such that when they get to this Line Integral introduction video, they are seeing the concept for the first time and usually do not yet have a well-developed sense of mathematical intuition about it nor the concepts leading up to it.

Is it possible to parametrize a reverse path?

It may work for the points where t = ‘a’ and t = ‘b’, but that is obvious, since at ‘t’ = ‘a’ or ‘b’, t = (a + b)/2 will be satisfied. This is just as logical as saying 2 terms are in an AP. And even if they are (which is not possible), there is no progression taking place, since d = 0.

How to make an integral move in the reverse direction?

Therefore, in order to make the integral move in the reverse direction you have to change how the integral interprets the end points (if that makes sense). Comment on blahdee327’s post “I think the key thing to realize is that T is what…” Posted 9 years ago.