Problem 330 - Calculus Volume 3

Question

For the following exercises, without using Stokes' theorem, calculate directly both the flux of \(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}\) over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. \(\mathbf{F}(x, y, z)=(x+2 z) \mathbf{i}+(y-x) \mathbf{j}+(z-y) \mathbf{k} ; S\) is a triangular region with vertices \((3,0,0),(0,3 / 2,0)\), and (0, 0,3)

Step-by-Step Solution

Answer

Flux and circulation integral are calculated by direct evaluation of the curl on the surface and line integral along the boundary, respectively.

Step 1: Provide Context

We need to calculate two things directly: the flux of \(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}\) over a surface and the circulation integral around its boundary. The given vector field is \(\mathbf{F}(x, y, z)=(x+2z) \mathbf{i}+(y-x) \mathbf{j}+(z-y) \mathbf{k}\). The surface \(S\) is a triangular region with vertices \((3,0,0),(0,3 / 2,0)\), and \((0, 0,3)\).

Step 2: Calculate \(\operatorname{curl} \mathbf{F}\)

To find \(\operatorname{curl} \mathbf{F}\), we use the formula for curl: \(abla \times \mathbf{F} \). Calculate the partial derivatives to find:\[\operatorname{curl} \mathbf{F} = \left(\frac{\partial}{\partial y}(z-y) - \frac{\partial}{\partial z}(y-x)\right)\mathbf{i} + \left(\frac{\partial}{\partial z}(x+2z) - \frac{\partial}{\partial x}(z-y)\right)\mathbf{j} + \left(\frac{\partial}{\partial x}(y-x) - \frac{\partial}{\partial y}(x+2z)\right)\mathbf{k}\]Solving these, we get \(\operatorname{curl} \mathbf{F} = \mathbf{i} + 3\mathbf{k} \).

Step 3: Define the Surface \(S\) and its Normal Vector \(\mathbf{N}\)

The surface \(S\) is a triangle in the first octant. We can obtain its equation using the vertices. First, find the vectors along the edges: \((0, 3/2, 0) - (3, 0, 0)\) and \((0, 0, 3) - (3, 0, 0)\), which simplifies to \((-3, 3/2, 0)\) and \((-3, 0, 3)\). The normal vector \(\mathbf{N}\) can be found by taking the cross product of these edge vectors.

Step 4: Compute the Cross Product and Normalize

The cross product of \((-3, 3/2, 0)\) and \((-3, 0, 3)\) is calculated as follows:\[\mathbf{N} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -3 & 3/2 & 0 \ -3 & 0 & 3 \end{array} \right| = (9/2, 9, 9/2)\]Normalize this to get a unit normal vector: \(\hat{\mathbf{N}} = \frac{1}{\sqrt{(9/2)^2 + 9^2 + (9/2)^2}}(9/2, 9, 9/2)\).

Step 5: Calculate the Flux of \(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}\)

The flux of \(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}\) is given by the integral \(\iint_S (\operatorname{curl} \mathbf{F}) \cdot \mathbf{N} \, dS\). Here, substitute \(\operatorname{curl} \mathbf{F} = \mathbf{i} + 3\mathbf{k}\) and the unit normal vector into the dot product. Since finding an explicit \(dS\) is complex, simplify computations by direct geometric evaluation, finding flux as a scalar multiple of the triangle's area.

Step 6: Evaluate the Circulation Integral Directly

To find the circulation integral \(\oint_C \mathbf{F} \cdot d\mathbf{r}\), parameterize the edges of the triangular boundary using lines between each pair of vertices. Compute \(\mathbf{F}\) dotted with the differential element \(d\mathbf{r}\) along each path and integrate over each interval. The total circulation is the sum over these edges.

Key Concepts

Flux

In vector calculus, **flux** describes how much of a vector field „flows" through a given surface. Imagine it as how wind might pass through a net. In more technical terms, the flux of a vector field \( \mathbf{F} \) through a surface \( S \) is calculated using a surface integral.

The expression \( \iint_S \operatorname{curl} \mathbf{F} \cdot \mathbf{N} \, dS \) represents this idea, with \( \operatorname{curl} \mathbf{F} \) being the rotational aspect of \( \mathbf{F} \), and \( \mathbf{N} \) the normal vector to the surface. The dot product \( \operatorname{curl} \mathbf{F} \cdot \mathbf{N} \) emphasizes how aligned the curl is with the normal, reflecting how much of the rotation penetrates the surface.

In our problem, once \( \operatorname{curl} \mathbf{F} \) and the normal vector are calculated, flux is determined by evaluating this integral directly. However, due to geometric complexity, simplification is achieved by noting how the scalar multiplication correlates with the surface area of the triangular region.

Circulation

**Circulation** in vector calculus measures the aggregated sum of a vector field along a closed loop. Think of it like walking on a track and measuring how much the wind pushes you forward. Mathematically, the circulation integral is written as \( \oint_C \mathbf{F} \cdot d\mathbf{r} \), where \( C \) is the closed curve, and \( d\mathbf{r} \) represents an infinitesimal path along the loop.

Given the triangular region in the exercise, we focus on its boundary path. The goal is to compute the integral for the vector field \( \mathbf{F} \) dot \( d\mathbf{r} \) along the triangle's edges. Each edge is parameterized, and the integrals along these edges are summed to find total circulation.

This approach helps provide insight into how much of the vector field is "circulating" around the boundary. It's an intuitive measure, akin to measuring swirl or rotation within the enclosed space.

Curl

When dealing with vector fields, the **curl** provides an insight into the field's rotational characteristics. It acts as a measure of how the field circles around a point. In simple terms, if a vector field resembles a whirlpool, the curl indicates the strength and direction of this swirling motion.

For any vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is expressed as \( abla \times \mathbf{F} \), resulting in another vector. The formula is given by: \[abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\] This computation requires evaluating partial derivatives of the field components, and it helps in visualizing how the field "twists" in space.

In our exercise, the curl \( \operatorname{curl} \mathbf{F} = \mathbf{i} + 3\mathbf{k} \) was derived, signifying a swirling component of the vector field that plays a crucial role in both the flux and circulation evaluations.