Question

Use Stokes' Theorem to write the circulation form of Green's Theorem in the $yz$ -plane.

Short answer

Answer: The circulation form of Green's Theorem in the $yz$-plane using Stokes' Theorem is given by ${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r}=0.$ This means the line integral around any simple closed curve $C$ in the $yz$-plane equals zero.

Step-by-step solution

Define the vector field

Define a vector field $\overrightarrow{F}$ such that it lies entirely in the $yz$-plane. A possible vector field for this can be given by $$\overrightarrow{F}(x,y,z)=⟨0,y,z⟩.$$ Note that the $x$ component is zero because $\overrightarrow{F}$ lies only in the $yz$-plane.

Compute the curl of the vector field

The curl of the vector field $\overrightarrow{F}$ is given by the following formula: $$\mathrm{curl}\overrightarrow{F}=\mathrm{\nabla }\times \overrightarrow{F}=⟨\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}⟩\times ⟨0,y,z⟩.$$ Calculate the curl as follows: $$\text{Extra left or missing right}$$

Apply Stokes' Theorem

Stokes' Theorem states $${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r}={\iint }_S(\mathrm{curl}\overrightarrow{F})\cdot d\overrightarrow{S},$$ where $C$ is the simple closed curve bounding the surface $S$. Since the curl of the vector field is $\mathrm{curl}\overrightarrow{F}=⟨0,0,0⟩$, applying Stokes' Theorem gives $${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r}={\iint }_S⟨0,0,0⟩\cdot d\overrightarrow{S}=0.$$

Conclude the circulation form of Green's Theorem

As a result, the circulation form of Green's Theorem in the $yz$-plane using Stokes' Theorem gives us $${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r}=0.$$ This indicates that the line integral around any simple closed curve $C$ in the $yz$-plane equals zero.

Key concepts

Circulation form of Green's Theorem

Green's Theorem relates a line integral around a simple closed curve in a plane to a double integral over the plane region it encloses. The circulation form of Green's Theorem is particularly interesting because it helps us connect the line integral of a vector field $\overrightarrow{F}$ around a closed curve $C$ to the curl of $\overrightarrow{F}$ within the region $D$ bounded by $C$. Simply put, it tells us that the total 'circulation' around the boundary of $D$ can be determined by summing up the tiny 'whirlpools' or curls of $\overrightarrow{F}$ inside $D$.

This concept is elegantly captured by the equation:
$${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r}={\iint }_D(\mathrm{curl}\overrightarrow{F})\cdot d\overrightarrow{S},$$
where ${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r}$ represents the circulation of $\overrightarrow{F}$ around $C$, and ${\iint }_D(\mathrm{curl}\overrightarrow{F})\cdot d\overrightarrow{S}$ represents the sum of the curls of $\overrightarrow{F}$ inside $D$. The $D$ herein would be the region in the $yz$ -plane enclosed by the curve for this particular application of Stokes' Theorem. When the curl of $\overrightarrow{F}$ is zero, as stated in the provided solution, it implies that there is no 'circulation' or rotational component of $\overrightarrow{F}$ inside region $D$ and hence the line integral of $\overrightarrow{F}$ around $C$ will also be zero.

Curl of a vector field

In vector calculus, the curl of a vector field is a vector that represents the infinitesimal rotation of the field at a given point. To visualize this, imagine the vector field as a flow of water and the curl as the rotation or swirl of water at any point. Mathematically, the curl is calculated using the cross product of the del operator ($abla$) with the vector field ($\overrightarrow{F}$).

The curl is defined as:$$\mathrm{curl}\overrightarrow{F}=abla\times \overrightarrow{F},$$
and for a three-dimensional field $\overrightarrow{F}=⟨F_x,F_y,F_z⟩$, this is computed as:$$\mathrm{curl}\overrightarrow{F}=⟨\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}⟩.$$
From the exercise provided, the vector field lies in the $yz$ -plane, thus eliminating any $x$ component, making the curl zero. This result profoundly informs us that the vector field $\overrightarrow{F}$ does not have any tendency to cause rotation around any axis within the specified plane, which aligns with the conclusion that the circulation around any closed curve in the $yz$ -plane is zero.

Line integral

A line integral is an integral where the function to be integrated is evaluated along a curve. It comes in several flavors, depending on the nature of the function being integrated. For vector fields, the line integral is often used to measure the 'work' done by the field in moving a particle along the curve. It is a fundamental tool in vector calculus, with applications ranging from physics to engineering.

The line integral of a vector field $\overrightarrow{F}$ along a curve $C$ is denoted by:$${\oint }_C\overrightarrow{F}\cdot d\overrightarrow{r},$$
where $\overrightarrow{F}\cdot d\overrightarrow{r}$ represents the dot product of the vector field and the differential element of the curve. Intuitively, this calculates how much of the vector field is pointing along the direction of the curve at each point, then adds these up over the path.

In the context of Green's Theorem and Stokes' Theorem, the line integral captures the total circulation of the vector field along the bounding curve. If the vector field's curl is zero, as demonstrated in the example, the line integral along any closed path in the plane of interest will also evaluate to zero, indicating a lack of rotational motion in the field along that path.

Vector calculus

Vector calculus is the branch of mathematics that deals with the differentiation and integration of vector fields, often in three-dimensional Euclidean space. It provides the language and tools for describing and analyzing the dynamics of physical fields, such as velocity fields in fluid dynamics or electromagnetic fields. The concepts of gradient, divergence, curl, and integral theorems are the pillars of vector calculus.

Vector calculus plays a crucial role in physics and engineering. It allows us to take a complicated physical situation and break it down into simpler, more manageable pieces. For instance, using the gradient, we can find the direction of the steepest ascent in a scalar field. The divergence gives us a measure of a vector field's tendency to diverge from or converge to a point. In our context, the curl determines the rotational tendencies of a field, and integrals allow us to sum up these local properties over an area or along a path. Stokes' Theorem is a perfect example of how vector calculus helps us connect local properties of a field (the curl) to global properties (the circulation around a boundary).

By understanding these fundamental concepts, one gains the ability to analyze and solve complex physical problems, such as predicting the behavior of fluids or the impact of forces on objects, which forms the essence of many technological and scientific advancements.