Determine whether the statement is true or false. Vector field 𝐅=⟨3 x^2, 1⟩is a gradient field f

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Determine whether the statement is true or false. Vector...

Determine whether the statement is true or false.

Vector field $\mathbf{F}=\left\langle 3 x^{2}, 1\right\rangle$ is a gradient field for both $\phi_{1}(x, y)=x^{3}+y$ and $\phi_{2}(x, y)=y+x^{3}+100.$

Key Concepts

Gradient Field

A gradient field is a vector field that can be expressed as the gradient of some scalar function. This means that each component of the vector field corresponds to the partial derivative of a potential function with respect to one of the variables, making the field conservative and ensuring that the line integral between any two points depends only on the endpoints.

Potential Function

A potential function is a scalar function whose gradient produces a given vector field. In the context of conservative fields, any potential function will generate the same gradient field, except they may differ by a constant. This property is crucial because it implies that while multiple functions can serve as potential functions, they are equivalent modulo an additive constant.

Conservative Vector Field

A conservative vector field is one where the work done along a path depends only on the start and end points. Such fields are associated with potential functions and satisfy certain conditions like having a zero curl in simply connected regions, ensuring that the field can be represented as the gradient of some scalar potential.

Additive Constant in Potential Functions

The potential function corresponding to a gradient field is not unique; any two potential functions for the same gradient field differ only by a constant. This means that if a function ? is a potential function for a vector field, then ? plus any constant will yield the same gradient, which is why both expressions given in the problem are valid potential functions.

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