Prove: If 𝐅(x, y, z)=f(x, y, z) 𝐢+g(x, y, z) 𝐣+h(x, y, z) 𝐤 is a conservative field and f, g, an

step 1 Procum 31 f of x y and z is equal to partial f partial x partial x g of x y and z is equal to pa

Prove: If 𝐅(x, y, z)=f(x, y, z) 𝐢+g(x, y, z) 𝐣+h(x, y, z) 𝐤...

Prove: If $$ \mathbf{F}(x, y, z)=f(x, y, z) \mathbf{i}+g(x, y, z) \mathbf{j}+h(x, y, z) \mathbf{k} $$ is a conservative field and $f, g,$ and $h$ are continuous and have continuous first partial derivatives in a region, then $$ \frac{\partial f}{\partial y}=\frac{\partial g}{\partial x}, \quad \frac{\partial f}{\partial z}=\frac{\partial h}{\partial x}, \quad \frac{\partial g}{\partial z}=\frac{\partial h}{\partial y} $$ in the region.

Prove: If
$$
\mathbf{F}(x, y, z)=f(x, y, z) \mathbf{i}+g(x, y, z) \mathbf{j}+h(x, y, z) \mathbf{k}
$$
is a conservative field and $f, g,$ and $h$ are continuous and have continuous first partial derivatives in a region, then
$$
\frac{\partial f}{\partial y}=\frac{\partial g}{\partial x}, \quad \frac{\partial f}{\partial z}=\frac{\partial h}{\partial x}, \quad \frac{\partial g}{\partial z}=\frac{\partial h}{\partial y}
$$
in the region.

Key Concepts

Conservative Vector Field

A conservative vector field is one that can be expressed as the gradient of a scalar potential function. This property implies that the work done by the field along any closed path is zero and that the field is path-independent, which is crucial in many physical applications such as energy conservation.

Potential Function

A potential function is a scalar function whose gradient yields the conservative vector field. The existence of such a function provides a way to simplify the analysis of the field, as many properties of the field can be inferred from the potential function, including its behavior and the relationships between its partial derivatives.

Gradient Fields and Mixed Partial Derivatives

When a vector field is conservative and its components have continuous first partial derivatives, the potential function associated with the field will have continuous second partial derivatives. By Clairaut's (or Schwarz's) theorem, this guarantees that the mixed partial derivatives are equal, leading directly to the relationships between the partial derivatives of the vector field’s components.

Curl and Irrotational Conditions

The curl of a vector field measures the tendency of the field to induce rotation. A conservative field, being expressible as a gradient, has a curl of zero (it is irrotational) in regions where the potential function is well-defined. This zero curl condition is equivalent to the equalities of the mixed partial derivatives and is used to verify the conservativeness of the field.

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