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How do line integrals work?

a) How line integrals arise. The figure on the left shows a force F being applied over a displacement r. Work is force times distance, but only the component of the force in the direction of the displacement does any work. So, work = |F| cos |r| = F · r.

How do you parameterize a line integral?

1:37 12:32 Suggested clip Line integrals: parametrization independence | MIT 18.02SC YouTubeStart of suggested client of suggested clip Line integrals: parametrization independence | MIT 18.02SC

What does a line integral mean?

A line integral (sometimes called a path integral) is the integral of some function along a curve. One can integrate a scalar-valued function along a curve, obtaining for example, the mass of a wire from its density. One can also integrate a certain type of vector-valued functions along a curve.

What is DS in line integral?

The line integral of f(x, y) along C is denoted by: The differential element is DS. This is the fact that we are moving along the curve, C, instead DX for the x-axis, or Dy for the y-axis. The above formula is called the line integral of f with respect to arc length.

Is work a line integral?

a) How line integrals arise. The figure on the left shows a force F being applied over a displacement r. Work is force times distance, but only the component of the force in the direction of the displacement does any work. So, work = |F| cos |r| = F · r.

What does a line integral do?

A line integral (sometimes called a path integral) is the integral of some function along a curve. One can integrate a scalar-valued function along a curve, obtaining for example, the mass of a wire from its density. One can also integrate a certain type of vector-valued functions along a curve.

Can a line integral 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.

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