Suppose that ∇f(x, y, z)=M(x, y, z) 𝐢+N(x, y, z) 𝐣+ P(x, y, z) 𝐤, where M, N, and P have continu

step 1 In this problem, as we know that this value is equal to f xI plus fy, j plus fz k. This means th

Suppose that ∇f(x, y, z)=M(x, y, z) 𝐢+N(x, y, z) 𝐣+ P(x, y,...

Suppose that $\nabla f(x, y, z)=M(x, y, z) \mathbf{i}+N(x, y, z) \mathbf{j}+$ $P(x, y, z) \mathbf{k},$ where $M, N,$ and $P$ have continuous first-order partial derivatives in an open set $D$. Prove that $$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}, \quad \frac{\partial M}{\partial z}=\frac{\partial P}{\partial x}, \quad \frac{\partial N}{\partial z}=\frac{\partial P}{\partial y}$$ in $D .$ Hint: Use Theorem $12.3 \mathrm{C}$ on $f$.

Key Concepts

Gradient Vector

A gradient vector is the vector of partial derivatives of a scalar function. It points in the direction of the greatest rate of increase of the function and its components represent the rates of change along the coordinate axes. In the context of potential functions, the gradient vector is used to define conservative fields, connecting scalar functions with the vector fields derived from them.

Conservative Vector Fields

A conservative vector field is one that can be expressed as the gradient of a scalar potential function. This implies that the integral of the vector field over any path only depends on the endpoints, and that the field has no rotational component (i.e., its curl vanishes). This concept is important because it allows us to make conclusions about the existence of potential functions and to use properties of partial derivatives to prove equality relations among the components of the vector field.

Equality of Mixed Partial Derivatives

The equality of mixed partial derivatives, also known as Clairaut's theorem, states that if the second-order partial derivatives of a function are continuous, then the mixed partial derivatives are equal regardless of the order of differentiation. This principle is crucial for vector calculus when proving that the corresponding components of a gradient vector field satisfy the equalities like ?M/?y = ?N/?x, ensuring the consistency of the gradient field when derived from a potential function.

Clairaut's Theorem

Clairaut's Theorem provides conditions under which the mixed partial derivatives of a function are equal. This theorem is key to understanding the structure of gradient fields with continuous partial derivatives, as it guarantees that the order of differentiation does not matter. This property is used to establish necessary relationships among the components of a gradient when the function is sufficiently smooth, ensuring the vector field is conservative.

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