Determine if the vector field is conservative. 𝐅(x, y)=(1)/(√(x^2+y^2))(x 𝐢+y 𝐣)

Name: Problem 32, Chapter 15: Calculus
Uploaded: 2021-10-07T11:08:57-08:00
Duration: 2 min 7 s
Channel: Harshita Goel
Description: Determine if the vector field is conservative. 𝐅(x, y)=(1)/(√(x^2+y^2))(x 𝐢+y 𝐣)

step 1 In this problem of vector field, we have to determine if the vector field is conservative or not

Determine if the vector field is conservative. 𝐅(x,...

Key Concepts

Conservative Vector Field

A conservative vector field is one that can be expressed as the gradient of a scalar potential function. In such fields, the work done along any path between two points is independent of the chosen path, meaning that the line integral depends only on the endpoints.

Potential Function

A potential function is a scalar function whose gradient equals the given vector field. The existence of a potential function for a field confirms its conservativeness, and it can be used to simplify the evaluation of line integrals by reducing them to differences in the potential function at the endpoints.

Curl Test for Conservativeness

For a vector field defined on a simply-connected region, a common method to check for conservativeness is to calculate its curl. If the curl of the vector field is zero everywhere in the region, then the field is conservative. This condition is a necessary and, in the context of simply-connected domains, sufficient condition for the existence of a potential function.

Simply-Connected Region

A simply-connected region is one in which every closed loop can be continuously contracted to a single point without exiting the region. The property of simple connectivity is important because, while a zero curl is necessary for a field to be conservative, in domains that are not simply-connected there may be exceptions where a non-conservative field still has zero curl.

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