Problem 37 - Calculus Early Transcendentals

Question

Let \(k\) be a constant, \(\mathbf{F}=\mathbf{F}(x, y, z), \mathbf{G}=\mathbf{G}(x, y, z),\) and \(\phi=\phi(x, y, z) .\) Prove the following identities, assuming that all derivatives involved exist and are continuous. $$ \operatorname{div}(\operatorname{curl} \mathbf{F})=0 $$

Step-by-Step Solution

Answer

The divergence of the curl of any vector field is always zero.

Step 1: State the identity

For any vector field \(\mathbf{F}\) with continuous second partial derivatives, \(\nabla \cdot (\nabla \times \mathbf{F}) = 0\).

Step 2: Prove by direct computation

If \(\nabla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\right)\mathbf{i} + \cdots\), then taking divergence and applying equality of mixed partials (Clairaut's theorem), all terms cancel pairwise, giving 0. \(\square\)

Key Concepts

Divergence

Divergence is a fundamental operator in vector calculus that measures how much a vector field spreads out or converges at a given point. To find the divergence of a vector field \( \mathbf{A} = (A_x, A_y, A_z) \), we calculate the dot product of the del operator \( abla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \) with \( \mathbf{A} \): \[ \operatorname{div} \mathbf{A} = abla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z} \] This scalar value produced, called the divergence, intuitively tells us if a point acts as a source (positive divergence) or a sink (negative divergence). If the divergence is zero, it implies that the vector field neither spreads out nor converges, suggesting an incompressible flow at that point.

Curl

The curl measures the rotation or swirling strength of a vector field around a point. It is particularly useful in studying fluid dynamics and electromagnetism. To compute the curl of a vector field \( \mathbf{F} = (F_x, F_y, F_z) \), we use the cross product of the del operator \( abla \) with the vector field: \[ \operatorname{curl} \mathbf{F} = abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \] This results in another vector, indicating the axis and strength of rotation at each point. A curl of zero in a field can be interpreted as the absence of rotational behavior, making the field irrotational.

Vector Field Identities

In vector calculus, several key identities simplify the complexity of analyzing vector fields. One central theorem is that the divergence of the curl of any vector field \( \mathbf{F} \) is always zero: \[ \operatorname{div}(\operatorname{curl} \mathbf{F}) = 0 \] This identity encapsulates a fundamental property of vector fields, signifying that rotation in a field does not contribute to any net outflow. This is essential when proving results in physics, such as electromagnetics, ensuring conservation laws and symmetry principles hold up under examination. These identities streamline the calculations we often undertake in fields like fluid dynamics and help verify solutions in mathematical physics models.