# What Is A Conservative Vector Field

What Is A Conservative Vector Field – Determine if a vector field is conservative Find the curve of a vector field Find the difference of a vector field

4 Note: Although a vector field consists of infinitely many vectors, you can get a good idea of ​​what a vector field looks like by drawing a few representative vectors F(x, y) whose initial points are (x, y).

## What Is A Conservative Vector Field

If div F=0, then F is said to be invariant. Note: Diffraction measures the rate of particle flux per unit volume at a point.

### Solution: Gradient Divergence And Curl Vectors

*We created a dot product to find the curve in the section. This method uses determinants Since this is a cross product, you should end up with a VECTOR field. Otherwise, if curl F=0, we say that F is non-rotating

A vector field is conservative if curl F(x, y, z)=0 When a vector field passes the conservative test, you can find the potential function by taking 3 separate integrals.

25 Applications Velocity fields (eg a wheel rotating on an axis – the further a point is from the axis, the greater its velocity Gravitational fields=attractive force acting on a piece of mass, i.e. Newton’s law of gravity (eg centripetal force field ) Electric fields or Coulomb’s law = force acting on a charged particle

26 Contour maps: the gradient points in the steepest direction above the surface (the contour field and the contour map must be orthogonal to each other)

Understanding the Fundamental Theorem on Line Integrals Understanding the concept of path independence Understanding the concept of conservation of energy

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Consider the vector field F. It is conservative – so the circulation of all simple closed curves (except those in the zero line) is zero. However, visually this seems absurd; Consider curve C which is a circle of radius 4 centered at the origin. If F is a force field, one definition of momentum is the work done on a particle as it travels a given path. The infinite amount of such work as the dot product of the vector field and the tangent vector at any point, multiplied by the infinite length of the arc: (F point T)ds – that is, the work done along. difficulty

By orienting the arch counterclockwise using the parametrization, it becomes clear that the vector field is always positively related to the tangent vector of the arch – and therefore NO NEGATIVE WORK IS DONE anywhere along the curve. However, the rotation around the circle is [zero][3]. By the same integral, the work done in the [second square][4] is negative.

## Determine Whether The Vector Field Is Conservative. If It Is

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My friends say vector field is conservative. But I don’t think so, the reason is that if you draw a closed curve on a vector field and add all the line integrals, I believe you will get a value greater than zero. For a vector field to be conservative, the integral must be zero.

Am I right that the vector field is not conservative. i Therefore, in part c of the question, none of the above.

You are right. To see that the vector field is not conservative, it will suffice to show a loop that has a non-zero line integral. You can easily do this with this vector field by starting at the origin, moving right along the x-axis, moving north, then back to the y-axis, and back to the origin from there. All components of this path will have an integral of \$0\$, except the second one, which has a positive integral.

### Independence Of Path And Conservative Vector Fields

It seems to me that the \$y\$ part of the vector field corresponds to \$x\$, ie. the vector field is somewhat similar to \$0dx+xdy\$.

It does not satisfy any of (a)-(c) because the necessary condition for the vector field \$f (x, y)dx+g (x, y)dy \$ to satisfy these conditions is \$frac =frac \$ is , which is not the case here (\$0not 1\$).

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## File:line Integral To Prove The Relation Between Path Independence And Conservative Vector Field

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