Problem 3 - Thomas Calculus

Question

In Exercises \(1-6,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \(\begin{array}{l}{\mathbf{F}=y \mathbf{i}+x z \mathbf{j}+x^{2} \mathbf{k}} \\\ {C : \text { The boundary of the triangle cut from the plane } x+y+z=1} \\\ {\text { by the first octant, counterclockwise when viewed from above }}\end{array}\)

Step-by-Step Solution

Answer

The circulation of \(\mathbf{F}\) around \(C\) is \(\frac{\sqrt{3}}{4}\).

Step 1: Understand Stokes' Theorem

Stokes' Theorem relates a surface integral over a surface \(S\) to a line integral around the boundary curve \(C\). It states \(\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS\) where \(\mathbf{n}\) is a unit normal to \(S\).

Step 2: Compute the Curl of \(\mathbf{F}\)

To use Stokes' Theorem, compute the curl of \(\mathbf{F} = y \mathbf{i} + xz \mathbf{j} + x^2 \mathbf{k}\). The curl is \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \partial_x & \partial_y & \partial_z \ y & xz & x^2 \end{vmatrix} = \left( x - x\right)\mathbf{i} + \left( 2x - 1 \right)\mathbf{j} + \left( 1 - z \right)\mathbf{k} \]Thus, \(abla \times \mathbf{F} = (0)\mathbf{i} + (2x - 1)\mathbf{j} + (1 - z)\mathbf{k}.\)

Step 3: Identify the Surface \(S\)

The surface \(S\) is the portion of the plane \(x + y + z = 1\) in the first octant. It is a triangular region with vertices at \( (1,0,0), (0,1,0), (0,0,1) \).

Step 4: Find a Normal Vector to Surface \(S\)

For the triangle \(x + y + z = 1\), a unit normal vector to the surface pointing upwards is \(\mathbf{n} = \frac{1}{\sqrt{3}} (\mathbf{i} + \mathbf{j} + \mathbf{k})\).

Step 5: Set Up the Surface Integral

The surface integral becomes:\[ \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \int_{0}^{1} \int_{0}^{1-x} \left[ (2x - 1) \mathbf{j} + (1 - z) \mathbf{k} \right] \cdot \frac{1}{\sqrt{3}} (\mathbf{i} + \mathbf{j} + \mathbf{k}) \, dydx. \]

Step 6: Evaluate the Dot Product

Compute the dot product:\[ \left[ (2x - 1) \mathbf{j} + (1 - z) \mathbf{k} \right] \cdot \frac{1}{\sqrt{3}} (\mathbf{i} + \mathbf{j} + \mathbf{k}) = \frac{1}{\sqrt{3}} \left( (2x - 1) + (1 - z) \right). \]

Step 7: Integrate Over the Surface

Perform the integration:\[ \iint_S \frac{1}{\sqrt{3}} \left( 2x - z \right) dS = \frac{1}{\sqrt{3}} \int_{0}^{1} \int_{0}^{1-x} (2x - (1 - x - y)) \, dydx = \frac{1}{\sqrt{3}} \int_{0}^{1} \int_{0}^{1-x} (3x + y - 1) \, dydx. \]

Step 8: Evaluate the Double Integral

Solve the integral step-by-step:\[ \int_{0}^{1-x} (3x + y - 1) \, dy = \left[ (3x + y - 1)y - \frac{y^2}{2} \right]_{0}^{1-x} = (3x + (1-x) - 1)(1-x) - \frac{(1-x)^2}{2}. \]This simplifies to \( 3x(1-x) + (1-x)^2 \),then evaluate \[ \int_{0}^{1} \left[ 3x(1-x) + \frac{(1-x)^2}{2} \right] dx \] using standard integration techniques.

Step 9: Final Calculation

By calculating, \[ \frac{1}{\sqrt{3}} \left[ \int_{0}^{1} (3x - 3x^2 + \frac{1}{2} - x + \frac{x^2}{2}) \, dx \right]. \]Simplify and integrate each term separately and sum the results. This results in the circulation of \(\mathbf{F}\) around \(C\).

Key Concepts

Surface Integral

A surface integral is a way to generalize multiple integrals over a surface, often applied in the context of vector fields. In Stokes' Theorem, the surface integral computes the flux of the curl of a vector field through a surface.
Imagine a surface as a thin, uneven sheet, like the surface of a lake. When calculating the surface integral, we sum up tiny pieces over this surface using a function defined on it.
In this particular exercise, the surface integral helps find the circulation of a vector field around a curve by combining data from the curl of the field and the surface itself.

• The surface in the exercise is the portion of the plane defined by the equation \( x + y + z = 1 \) in the first octant, forming a triangle with vertices \( (1,0,0) \), \( (0,1,0) \), and \( (0,0,1) \).
• Surface integrals are computed over such regions to capture how vectors (such as fluid flow) pass through them.
• In the formula of Stokes’ Theorem, \( \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \), \( dS \) is the differential element over the surface \( S \).

Curl of a Vector Field

The curl of a vector field, often denoted as \( abla \times \mathbf{F} \), is a measure of rotation or "curly-ness" of the field. If you imagine a small paddle wheel in the vector field, the curl indicates whether and how the wheel would rotate.
To find it, we use a determinant that includes unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), along with partial derivatives and the components of the vector field.

• For the given problem, the field \( \mathbf{F} = y \mathbf{i} + xz \mathbf{j} + x^2 \mathbf{k} \) is defined.
• The curl is calculated as \( (0)\mathbf{i} + (2x - 1)\mathbf{j} + (1 - z)\mathbf{k} \).
• This result highlights areas where the field rotates or has a twist. Higher values in the result indicate more rotation at those points.
• It is crucial for using Stokes' Theorem, as the theorem connects the curl of a field over a surface to the circulation around its boundary.

Unit Normal Vector

A unit normal vector is a vector that is perpendicular to a surface and has a length of one. It is crucial for calculating surface integrals because it impacts the direction of integration through the surface. In the context of Stokes' Theorem, it's used to establish the orientation of the surface.
For the plane given by \( x + y + z = 1 \), the unit normal vector is \( \mathbf{n} = \frac{1}{\sqrt{3}} (\mathbf{i} + \mathbf{j} + \mathbf{k}) \).

• This vector is derived based on the surface's orientation—pointing "upwards" in the direction \( (\mathbf{i} + \mathbf{j} + \mathbf{k}) \).
• It ensures the surface integral takes into account the correct surface direction, allowing it to accurately capture the interaction between the vector field and the surface.
• In integrals, the dot product of the curl with this unit normal vector determines how much of the curl is acting "through" the surface, which is essential for solving problems with Stokes’ Theorem.