In Problems 33-36, determine whether each vector field 𝐅 is a conservative vector field. 𝐅(x, y)

Name: Problem 33, Chapter 15: Calculus, Early Transcendentals
Uploaded: 2021-10-28T06:37:00-08:00
Duration: 1 min 59 s
Channel: Harshita Goel
Description: In Problems 33-36, determine whether each vector field 𝐅 is a conservative vector field. 𝐅(x, y)=x^2 𝐢+y^2 𝐣

step 1 In this problem of fundamental theorem of the line integral we have to determine whether each ve

In Problems 33-36, determine whether each vector field 𝐅 is...

Key Concepts

Simply-Connected Domain

A simply-connected domain is a region in which any closed path can be continuously contracted to a point without exiting the region. This property is crucial for the applicability of the partial derivative test; it ensures that a zero curl (i.e., irrotational field) is sufficient to guarantee that a vector field is conservative.

Partial Derivative Test

The partial derivative test is a technique used to determine if a vector field in two dimensions is conservative. For a vector field expressed as F = P i + Q j, the field is conservative if the mixed partial derivatives satisfy ?P/?y = ?Q/?x. This test provides a quick method for verifying the potential existence of a scalar potential function for the field.

Potential Function

A potential function is a scalar function whose gradient equals the given vector field. If a vector field F is conservative, then there exists a function f such that F = ?f. This function represents the potential energy in physical contexts and provides a useful method for evaluating work done by the field along any path.

Conservative Vector Field

A conservative vector field is one that can be derived from a scalar potential function. In such fields, the line integral between any two points depends solely on the end points, not on the path taken. This path independence is a hallmark of conservative fields and is intimately related to the concept of energy conservation in physics.

Gradient Field

A gradient field is a vector field that arises as the gradient of a scalar function. The defining property of a gradient field is that it is irrotational, meaning that its curl is zero in simply-connected regions. This characteristic is used to establish whether a given vector field is conservative.

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