Show that if the vector field 𝐅=P 𝐢+Q 𝐣+R 𝐤 is conservative and P, Q, R have continuous first-or

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Show that if the vector field 𝐅=P 𝐢+Q 𝐣+R 𝐤 is conservative...

Show that if the vector field $\mathbf{F}=P \mathbf{i}+Q \mathbf{j}+R \mathbf{k}$ is conservative and $P, Q, R$ have continuous first-order partial derivatives, then $$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x} \quad \frac{\partial P}{\partial z}=\frac{\partial R}{\partial x} \quad \frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}$$

Show that if the vector field $\mathbf{F}=P \mathbf{i}+Q \mathbf{j}+R \mathbf{k}$ is conservative and $P, Q, R$ have continuous first-order partial derivatives, then
$$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x} \quad \frac{\partial P}{\partial z}=\frac{\partial R}{\partial x} \quad \frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}$$

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 line integral between any two points is independent of the path taken. This property plays a key role in physics and engineering, particularly when dealing with force fields, as it implies the existence of energy conservation laws.

Potential Function

The potential function is a scalar function whose gradient gives the conservative vector field. Its existence allows computations involving work or energy to be simplified, as the value of the potential function at two points directly determines the net work done by the field. The relationship between the potential function and the vector field underpins many theoretical and practical applications.

Curl of a Vector Field

The curl is a vector operator that measures the rotation or 'twist' of a vector field. For a conservative field, the curl is zero, meaning there is no intrinsic rotation at any point in the field. This property is equivalent to the equality of certain mixed partial derivatives of the components of the vector field, which is essential in verifying a field’s conservativeness.

Clairaut's Theorem

Clairaut's theorem (also known as the symmetry of second derivatives) states that if the mixed partial derivatives of a function are continuous, then they are equal. This theorem justifies the equalities of mixed partial derivatives required in the problem, linking the mathematical conditions of conservative vector fields to the potential function's continuous second derivatives.

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