91Ó°ÊÓ

In vector calculus, the curl of a vector field provides information about the rotational properties of the field. Imagine a tiny paddlewheel placed in the vector field; the curl tells you how likely it is to spin. For a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is calculated as:\[abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k}\]In this exercise, we consider \( \mathbf{F}(x, y, z) = xy e^{z} \mathbf{i} + 0\mathbf{j} + yz e^{x} \mathbf{k} \)

• The \( \mathbf{i} \) component, \( \frac{\partial R}{\partial y} \), gives \( z e^{x} \) because \( R = yz e^{x} \) when differentiated with respect to \( y \).
• The \( \mathbf{j} \) component emerges from \( \frac{\partial P}{\partial z} \) which gives \( xy e^{z} \) minus \( \frac{\partial R}{\partial x} \) which is \( yz e^{x} \).
• The \( \mathbf{k} \) component simply becomes \( -x e^{z} \) as \( \frac{\partial P}{\partial y} = x e^{z} \) and the other term is zero.

The components form the curl \( abla \times \mathbf{F} = z e^{x} \mathbf{i} + (xy e^{z} - yz e^{x}) \mathbf{j} - x e^{z} \mathbf{k} \).

The divergence of a vector field is a measure of how much the field spreads out from a point, like how water might diverge from a source. For a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the divergence is found by summing these partial derivatives:\[abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\]In our exercise with \( \mathbf{F}(x, y, z) = xy e^{z} \mathbf{i} + 0\mathbf{j} + yz e^{x} \mathbf{k} \), we observe:

• \( \frac{\partial P}{\partial x} = y e^{z} \), due to the linear term in \( x \) in \( P = xy e^{z} \).
• \( \frac{\partial Q}{\partial y} = 0 \) since \( Q = 0 \) in the field.
• \( \frac{\partial R}{\partial z} = y e^{x} \) results from \( R = yz e^{x} \).

Adding these gives the divergence \( abla \cdot \mathbf{F} = y(e^{z} + e^{x}) \). This expression illustrates the spread at any point \( (x, y, z) \).

Partial derivatives are fundamental in vector calculus, especially when dealing with functions of multiple variables. They represent the rate of change of a function in one variable while keeping other variables constant. For a multivariable function \( f(x, y, z) \), the partial derivative with respect to \( x \), denoted \( \frac{\partial f}{\partial x} \), measures how \( f \) changes as \( x \) changes, while \( y \) and \( z \) remain unchanged.
In the context of vector fields, partial derivatives are used to compute both curl and divergence. Each component of the vector field is treated as a function of spatial variables, and partial derivatives help establish how these components interact over small changes in space:

• For the curl, we differentiate with respect to different pairs of variables to measure the rotation effects.
• For the divergence, partial derivatives allow us to assess how the vector field behaves as it diverges from or converges towards a point.

By applying partial derivatives to solve curl and divergence, we gain a deeper insight into the behavior of the vector field.