Problem 31 - Calculus Early Transcendentals

Question

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.

Step-by-Step Solution

Answer

The conditions are satisfied due to the symmetry of mixed partial derivatives in a conservative field.

Step 1: Understand the Definition of a Conservative Field

A vector field \( \mathbf{F}(x, y, z) \) is conservative if it can be expressed as the gradient of a scalar potential function \( \phi(x, y, z) \). This means \( \mathbf{F} = abla \phi \), which implies:\[ f = \frac{\partial \phi}{\partial x}, \quad g = \frac{\partial \phi}{\partial y}, \quad h = \frac{\partial \phi}{\partial z} \]

Step 2: Utilize Mixed Partial Derivative Property

The existence of \( \phi \) with continuous second-order partial derivatives leads to the equality of mixed partial derivatives on any order, which is the key property used in conservative fields. Specifically, \( \frac{\partial^2 \phi}{\partial x \partial y} = \frac{\partial^2 \phi}{\partial y \partial x} \), and similarly for the other pairs of mixed derivatives.

Step 3: Derive the First Condition

Given that \( f = \frac{\partial \phi}{\partial x} \) and \( g = \frac{\partial \phi}{\partial y} \), the mixed derivative equality \( \frac{\partial}{\partial y}(\frac{\partial \phi}{\partial x}) = \frac{\partial}{\partial x}(\frac{\partial \phi}{\partial y}) \) implies \( \frac{\partial f}{\partial y} = \frac{\partial g}{\partial x} \).

Step 4: Derive the Second Condition

Similarly, since \( f = \frac{\partial \phi}{\partial x} \) and \( h = \frac{\partial \phi}{\partial z} \), applying the mixed derivative equality \( \frac{\partial}{\partial z}(\frac{\partial \phi}{\partial x}) = \frac{\partial}{\partial x}(\frac{\partial \phi}{\partial z}) \) results in \( \frac{\partial f}{\partial z} = \frac{\partial h}{\partial x} \).

Step 5: Derive the Third Condition

Finally, using \( g = \frac{\partial \phi}{\partial y} \) and \( h = \frac{\partial \phi}{\partial z} \), the mixed derivative equality \( \frac{\partial}{\partial z}(\frac{\partial \phi}{\partial y}) = \frac{\partial}{\partial y}(\frac{\partial \phi}{\partial z}) \) ensures \( \frac{\partial g}{\partial z} = \frac{\partial h}{\partial y} \).

Step 6: Conclusion

These conditions solely depend on the symmetry of second-order partial derivatives, which are guaranteed under the continuity given in the problem statement. Thus, the conditions are satisfied for any conservative field in the specified region.

Key Concepts

Mixed Partial Derivatives

When we talk about mixed partial derivatives, we are exploring the interesting world of how functions change when we tweak one variable and then another. If you have a function that's nice and smooth, you can take partial derivatives in different orders, and usually, they turn out the same. This is called symmetry of mixed partial derivatives.

Imagine a surface with little wrinkles. The symmetry tells us that if you move along the surface by changing one variable and then another, you end up in the same spot as if you had changed the order of the variables. For a function \( \phi(x, y, z) \), this means things like \( \frac{\partial^2 \phi}{\partial x \partial y} = \frac{\partial^2 \phi}{\partial y \partial x} \).

Having continuous mixed partial derivatives is key in proving whether a vector field is conservative. It's like a litmus test for checking if our surface is well-behaved enough for these properties to hold true.

Scalar Potential Function

The scalar potential function is the hidden gem behind a conservative vector field. It gives us a simple way to describe the vector field using just one function, \( \phi(x, y, z) \). When a vector field \( \mathbf{F} \) is conservative, it means there is some magical scalar function from which the vector field can be derived.

In this context, the vector field \( \mathbf{F} \) is like the shadow on the ground, and the scalar function \( \phi \) is the object casting that shadow. The gradient of this scalar function points us in the direction of the vector field's flow, providing a bridge between the abstract potential function and the directions spelled out by \( \mathbf{F} \).

This connection means that once you master \( \phi \), you can easily compute the vector components: \( f = \frac{\partial \phi}{\partial x} \), \( g = \frac{\partial \phi}{\partial y} \), and \( h = \frac{\partial \phi}{\partial z} \). This relationship can greatly simplify solving problems involving vector fields.

Gradient

The gradient is a vector filled with the steepest ascent information from a scalar function. If you think about a hill, the gradient gives you the direction that takes you fastest to the top.

For a scalar function \( \phi(x, y, z) \), its gradient is represented as \( abla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right) \). Each component tells how quickly the function value changes when you stretch out along one of the coordinate axes.

What's fascinating about the gradient is that, in conservative vector fields, it's what defines the vector field. If \( \mathbf{F} \) is the gradient of \( \phi \), that makes \( \mathbf{F} \) conservative. This is why you can check for when a field is conservative by seeing if it can be presented as a gradient—if so, it's guaranteed to have those nice properties like symmetry of mixed partial derivatives.