# Path-Dependent Examples Of Vector Fields

This post categorized under Vector and posted on March 16th, 2020.

The vector field dlvf(xy) (y -x) is an example of a path-dependent vector field. This vector field represents clockwise circulation around the origin . It turns out such circulation is the key indicator of path-dependence (however the circulation may not be as obvious as it is in this example). In vector calculus a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent the choice of any path between two points does not change the value of the line integral.Path independence of the line integral is equivagraphict to the vector field being conservative. Vector fields provide an interesting way to look at the world. First a quick bit of background. A vector is a quangraphicy with magnitude and direction. A simple example is the velocity of a car that is traveling at 100 kmh in a Northerly direction.

Short answer They are fancy words for functions (usually in context of differential equations). Scalar fields takes a point in graphice and returns a number. Vector fields takes a point in graphice and returns a vector. Usually best understood in the c Vector fields let you visualize a function with a two-dimensional input and a two-dimensional output. You end up with well a field of vectors sitting at various points in two-dimensional graphice. See an example of how you can start to understand how the formula for a three-dimensional vector field relates to the way it looks. About Khan Academy Khan Academy offers practice exercises

For certain vector fields the amount of work required to move a particle from one point to another is dependent only on its initial and final positions not on the path it takes. Gravitational and electric fields are examples of such vector fields. This section will discuss the properties of these vector fields. Showing that unlike line integrals of scalar fields line integrals over vector fields are path direction dependent. If youre seeing this message it means were having trouble loading external resources on our website. If youre behind a web filter please make Example of taking a closed line integral of a conservative field Watch the next lesson httpswww.khanacademy.orgmathmultivariable-calculusline_integral

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