Problem 17 - Calculus Early Transcendentals

Question

Consider the vector field given by the formula $$ \mathbf{F}(x, y, z)=(x-z) \mathbf{i}+(y-x) \mathbf{j}+(z-x y) \mathbf{k} $$ (a) Use Stokes' Theorem to find the circulation around the triangle with vertices \(A(1,0,0), B(0,2,0),\) and \(C(0,0,1)\) oriented counterclockwise looking from the origin toward the first octant. (b) Find the circulation density of \(\mathbf{F}\) at the origin in the direction of \(\mathbf{k}\). (c) Find the unit vector \(\mathbf{n}\) such that the circulation density of \(\mathbf{F}\) at the origin is maximum in the direction of \(\mathbf{n} .\)

Step-by-Step Solution

Answer

(a) Circulation = \(-\frac{4}{\sqrt{6}}.\) (b) Circulation density in \(\mathbf{k}\) direction = -1. (c) Maximizing unit vector \(\mathbf{n} = \frac{1}{\sqrt{2}}(0\mathbf{i} - 1\mathbf{j} - 1\mathbf{k}).\)

Step 1: Parametrize the Triangle for Stokes' Theorem

To apply Stokes' Theorem, first parametrize the boundary of the triangle. The path is oriented counterclockwise: 1. From A(1,0,0) to B(0,2,0): \( \mathbf{r}_1(t) = (1-t)\mathbf{i} + 2t \mathbf{j} + 0 \mathbf{k}, \) where \( 0 \leq t \leq 1. \)2. From B(0,2,0) to C(0,0,1): \( \mathbf{r}_2(t) = 2(1-t) \mathbf{j} + t \mathbf{k}, \) where \( 0 \leq t \leq 1. \)3. From C(0,0,1) to A(1,0,0): \( \mathbf{r}_3(t) = t \mathbf{i} + (1-t)\mathbf{k}, \) where \( 0 \leq t \leq 1. \)

Step 2: Compute the Curl of the Vector Field

The curl of \( \mathbf{F} \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial F_k}{\partial y} - \frac{\partial F_j}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_i}{\partial z} - \frac{\partial F_k}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_j}{\partial x} - \frac{\partial F_i}{\partial y} \right) \mathbf{k}. \)Compute each component:- \( F_i = x-z, F_j = y-x, F_k = z-xy \)- \( \frac{\partial F_k}{\partial y} = -x, \frac{\partial F_j}{\partial z} = 0 \)- \( \frac{\partial F_i}{\partial z} = -1, \frac{\partial F_k}{\partial x} = -y \)- \( \frac{\partial F_j}{\partial x} = -1, \frac{\partial F_i}{\partial y} = 0 \)Thus, \( abla \times \mathbf{F} = -x \mathbf{i} + (y-1) \mathbf{j} - \mathbf{k}. \)

Step 3: Apply Stokes' Theorem

Stokes' Theorem states that \( \int_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S}, \) where \(C\) is the boundary curve and \(S\) is the surface enclosed by \(C\).For our triangular surface, find a normal vector \( \mathbf{n} \) of the triangle. The vertices give \( \mathbf{AB} = -\mathbf{i} + 2\mathbf{j}, \mathbf{AC} = -\mathbf{i} + \mathbf{k} \). Calculate \( \mathbf{n} = \mathbf{AB} \times \mathbf{AC} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}. \) Normalize \( \mathbf{n} \), so \( \mathbf{n} = \frac{1}{\sqrt{6}}(2\mathbf{i} + \mathbf{j} + \mathbf{k}). \)Compute \( \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = abla \times \mathbf{F} \cdot \mathbf{n} \cdot \text{Area} \). The area of the triangle is \( \frac{1}{2} \sqrt{6}. \) The circulation is \(-x\cdot\frac{2}{\sqrt{6}} + (y-1)\cdot\frac{1}{\sqrt{6}} - 1\cdot\frac{1}{\sqrt{6}}\), where at the center \(x = y = 0. \) Circulation is \(-\frac{2}{\sqrt{6}} - \frac{1 + 1}{\sqrt{6}} = -\frac{4}{\sqrt{6}}. \)

Step 4: Find Circulation Density at Origin in \(\mathbf{k}\) Direction

The circulation density at a point is simply the value of the curl at that point projected in a particular direction. At the origin \((0,0,0)\), the curl \( abla \times \mathbf{F} = 0\mathbf{i} - 1\mathbf{j} - 1\mathbf{k}. \) Thus, the circulation density in the \(\mathbf{k}\) direction is \(-1.\)

Step 5: Find Unit Vector Maximizing Circulation Density at Origin

The maximum circulation density at the origin will occur in the direction of the curl vector at the origin. Since \( abla \times \mathbf{F} = 0\mathbf{i} - 1\mathbf{j} - 1\mathbf{k} \) at (0,0,0), the unit vector is \( \mathbf{n} = \frac{1}{\sqrt{2}}(0\mathbf{i} - 1\mathbf{j} - 1\mathbf{k}). \)

Key Concepts

Vector Field

A vector field is a mathematical function that assigns a vector to each point in a space. Imagine it like a field of arrows. Each arrow represents the magnitude and direction of the vector at that particular point.
These fields can be visualized in two or three dimensions and they are extremely useful in physics and engineering. In the given exercise, the vector field is described by the function \( \mathbf{F}(x, y, z)=(x-z) \mathbf{i}+(y-x) \mathbf{j}+(z-x y) \mathbf{k} \). This means that at any point \((x, y, z)\), you plug these coordinates into the formula to find the corresponding vector.

• The \(i,j,k\) components here indicate the direction along the x, y, and z axes respectively.
• Such fields can represent anything from the flow of a fluid to electromagnetic forces.

Understanding vector fields is crucial for solving many types of problems in vector calculus, such as those involving Stokes' Theorem.

Curl of a Vector Field

The curl of a vector field gives us a measure of the rotation or 'twist' at a point in the field. Think of it like looking at how much a fluid 'swirls' around a point. Mathematically, it's another vector field that shows the axis of rotation and the rate of rotation at each point.
In our exercise, the curl is calculated using the components of the given vector field. To find the curl \((abla \times \mathbf{F})\), you do partial differentiation of the vector components according to a special formula:\[abla \times \mathbf{F} = \left( \frac{\partial F_k}{\partial y} - \frac{\partial F_j}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_i}{\partial z} - \frac{\partial F_k}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_j}{\partial x} - \frac{\partial F_i}{\partial y} \right) \mathbf{k}.\]

• In our specific case, the result was: \(-x \mathbf{i} + (y-1) \mathbf{j} - \mathbf{k}\),
• which provides the rotational tendency of the field at various points.

Comprehending the curl is essential in understanding phenomena described by Stokes' Theorem and other vector calculus concepts.

Circulation Density

Circulation density refers to the amount of 'spinning' or circulation of the vector field at certain points and in specific directions. It is an important concept as it helps to understand the rotational behavior of the field. At a particular point, this is directly influenced by the curl of the vector field.
In the exercise, the circulation density at the origin is analyzed, specifically in the \(\mathbf{k}\) direction. By projecting the curl at this point onto the \(\mathbf{k}\) direction, you obtain the circulation density: \(-1\) in this case.

• This result tells you how 'intense' the field's rotation is in the \(\mathbf{k}\) direction.
• For different directions, such as the maximum circulation density, you would look at the magnitude and direction of the curl vector itself.

Understanding how to calculate and interpret circulation density is pivotal for evaluating and working with vector fields in practice.