Simple line integral problem

In summary, the conversation discusses a question about evaluating line integrals around a curve and finding the gradient of a scalar. The first line integral results in 0 and the second one is calculated to be -π(rz0)^2. The problem then asks to use the gradient result to explain the answers to the line integrals. One person suggests using Green's theorem and treating the problem as a 2D one in the xy-plane. The other person mentions that the curl does not come out to -z0 and they are unable to link it to the gradient of the scalar. They ask for further elaboration on the first person's thoughts.
  • #1
I have a question which asked me to evalute the line integral around the curve x^2+y^2=r^2 (z=z0 (a constant)) of the following vectors:

(0, z^2, 2yz)
(yz^2, yx^2, xyz)

the first one I get as 0, and the second one I get as: -pi(r*z0)^2

Those answers I'm pretty sure are right

The next part of the problem asks to find grad(yz^2) which I calculate to be: (0,z^2, 2yz).

The problem then asks to use this result to explain the answers to the two line integrals in the first part of the question. Now the first integral is easy to explain...since it is the integral of the gradient of a scalar...hence a conservative field, hence the integral around a closed loop i.e. a circle, is 0!

But I can't seem to explain the 2nd integral, any ideas anyone?
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  • #2
Have you had Green's theorem? since z remains constant (z0) thoughout the integration, you can treat this as a problem in the xy-plane. What would Green's theorem tell you about that integral?

(Oh, by the way, it's easy to integrate xy over a disk with center at (0,0) isn't it?_
  • #3
well i thought about it being something to do with greens theorem...hence just do the curl of the vector over the area but the curl doesn't come out to -z0...i just can't link it to the grad of that scalar i gave. Do you care to elaborate on what you're thinking?

1. What is a simple line integral problem?

A simple line integral problem involves calculating the integral of a function along a straight line segment in a two-dimensional or three-dimensional space. It is used in various fields of science and mathematics to determine the area under a curve, work done by a force, or the length of a curve.

2. How do you solve a simple line integral problem?

To solve a simple line integral problem, you need to first determine the limits of integration, which are the starting and ending points of the line segment. Then, you need to use the appropriate formula to calculate the integral, which involves multiplying the function by the infinitesimal length of the line segment and integrating over the limits.

3. What are the applications of simple line integrals?

Simple line integrals have various applications in physics, engineering, and other fields. They are used to calculate work done by a force, electric and magnetic fields, fluid flow, and many other physical quantities. They are also used in vector calculus to solve problems involving surfaces and volumes.

4. Can you give an example of a simple line integral problem?

One example of a simple line integral problem is calculating the work done by a force along a straight path. If the force is represented by a vector field and the path is given by a parametric equation, the line integral can be used to determine the work done by the force along that path.

5. What are the differences between a simple line integral and a double integral?

A simple line integral is a one-dimensional integral, while a double integral is a two-dimensional integral. In a simple line integral, the limits of integration are points on a line, while in a double integral, the limits are regions in a plane. Additionally, a double integral involves integrating a function over a surface, while a simple line integral involves integrating a function over a line segment.

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