Common notation

A common notation that we will use in our discussions in the following weeks is the vector differential operator $\nabla$ (“del”) as

Definitions

We will define two quantities, a vector and a scalar, by using the outer product and inner product respectively.

Curl

We define curl of a vector field $F=Pi+Qj+Rk$ over $\mathbb{R}^3$ as

The expanding of $\text{curl}(F)$ is

Divergence

We define divergence of a vector field $F$ over $\mathbb{R}^3$ (or $\mathbb{R}^2$) as

The divergence of a gradient vector field, symbolized as $\nabla f$, is a concept that frequently comes up:

Following this, we introduce the Laplace operator, defined as:

The Laplace operator, also known as the Laplacian, plays a significant role in many areas of physics, such as in the study of heat conduction, electricity, and magnetism. It is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient of a function. In simple terms, it measures the difference between a function’s average value around a point and the function’s value at that point.

$\text{Div}(\text{curl(F)})$

We have the following theorem:

Theorem. If all of the component function of $F$ have a continuous second order partial differentiation, then we have

This theorem allows us to determine whether a vector field $F$ exists, such that $\text{curl}(F)=G$, by checking if $\text{div}(G)=0$.

Criteria for a Conservative Vector Field in $\mathbb{R}^3$ and the Meaning of Curl

Criteria

Similar to a vector field in $\mathbb{R}^2$, we use the first-order partial derivative to determine whether a vector field is conservative. Importantly, the first-order partial derivative of a conservative field corresponds to the second-order partial derivative of certain real functions.

Theorem. If $F$ is a vector field defined on all of $\mathbb{R}^3$ whose component functions have continuous partial derivatives and $\text{curl}(F)=0$, then $F$ is conservative vector field.

Meaning

By adding $0k$ to a plane vector field, we extend the definition of a plane vector field $F=Pi+Qj$ to a space vector field $F=Pi+Qj+0k$. The curl of $F$ is $0i+0j+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)k$. We observe that the coefficient of $k$ corresponds exactly to the function used in Green’s theorem.

Let’s use the following graphs to better understand what a curl measures.

Let’s examine $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$ at the origin, which is the center of each graph. This partial differential function $\frac{\partial P}{\partial y}$ measures the change in the $x$-coordinate of vectors along the $y$-axis. In Fig 1, the vectors near the origin change from $\downarrow$ to $\uparrow$ along the $y$-axis, where the $x$-coordinate is unchanged. As a result, $\frac{\partial P}{\partial y}=0$. However, in Fig 2, the vector transitions from $\rightarrow$ to $\leftarrow$, indicating that $\frac{\partial P}{\partial y}<0$.

On the other hand, $\frac{\partial Q}{\partial x}$ measures the change in the $y$-coordinate of vectors along the $x$-axis. In Fig 1, vectors near the origin switch from $\leftarrow$ to $\rightarrow$ along the $x$-axis, hence $\frac{\partial Q}{\partial x}=0$. In contrast, in Fig 2, the vectors switch from $\downarrow$ to $\uparrow$ along the $x$-axis, so $\frac{\partial Q}{\partial x}>0$.

Thus, we analyze that $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=0$ for Fig 1, and $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}>0$ for Fig 2. Therefore, we can intuitively suggest that the curl measures the degree of rotation in a vector field.

Connecting with Green’s theorem

The formula

indicates that on the right-hand side, we total the degree of rotation in a region $D$. On the other side, we measure the work of the vector field acting on the boundary of $D$.