Problem 5 - Advanced Engineering Mathematics

Question

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}=(2 z+x) \mathbf{i}+(y-z) \mathbf{j}+(x+y) \mathbf{k} ; C\) the triangle with vertices \((1,0,0),(0,1,0),(0,0,1)\)

Step-by-Step Solution

Answer

The value of the line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is 0.

Step 1: Identify the surface S with boundary C

To apply Stokes' theorem, we need to identify a surface \(S\) whose boundary is \(C\). In this case, \(S\) is the triangular portion of the plane \(x+y+z=1\) that encloses \(C\).

Step 2: Find the normal vector and parameterize the surface

For the triangle, the equation of the plane is \(x + y + z = 1\). From this, we find the normal vector \(\mathbf{n} = \mathbf{i} + \mathbf{j} + \mathbf{k}\). The surface can be parameterized using parameters \(u\) and \(v\) as \(\mathbf{r}(u,v) = u\mathbf{i} + v\mathbf{j} + (1-u-v)\mathbf{k}\), with \(0 \leq u \leq 1\) and \(0 \leq v \leq 1-u\).

Step 3: Find the curl of F

The curl of \(\mathbf{F}\) is found using the formula \(abla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\right)\mathbf{i} + \left(\frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}\right)\mathbf{j} + \left(\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right)\mathbf{k}\). For \(\mathbf{F} = (2z+x) \mathbf{i} + (y-z) \mathbf{j} + (x+y) \mathbf{k}\), the curl is \(abla \times \mathbf{F} = -2\mathbf{i} + 1\mathbf{j} + 1\mathbf{k}\).

Step 4: Evaluate the surface integral using Stokes' theorem

Stokes' theorem states that \(\oint_C \mathbf{F} \cdot d\mathbf{r} = \int \int_S (abla \times \mathbf{F}) \cdot d\mathbf{S}\). We have \(abla \times \mathbf{F} = -2\mathbf{i} + 1\mathbf{j} + 1\mathbf{k}\) and \(d\mathbf{S} = \mathbf{n} dudv = (\mathbf{i} + \mathbf{j} + \mathbf{k}) dudv\). The integral becomes \(\int \int_S (-2 \cdot 1 + 1 \cdot 1 + 1 \cdot 1) dudv\). This simplifies to \(\int \int_S 0 dudv\), which evaluates to 0 over the region as the integrand is 0.

Key Concepts

Surface Integrals

Surface integrals are a fascinating concept in vector calculus. They extend the idea of integrating over a space from one dimension to two dimensions. Here, instead of summing function values along a curve, we're summing them across a surface.
This is crucial for applications where you're examining vector fields across surfaces, such as computing the flux of a vector field through a surface. In the context of Stokes' Theorem, the surface integral considers the curl of a vector field over a surface and relates it to a line integral over the boundary of that surface.

• This relationship assists in simplifying the complexity of evaluating integrals over paths by converting them to integrals over the surfaces enclosed by these paths.
• In the given exercise, the surface integral over the triangular plane surface is determined using the surface bounded by the curve.
• The result of the integral being zero indicates the balancing forces within the planar boundary defined.

Understanding surface integrals is not only key to mastering Stokes' theorem but also aids in grasping more advanced topics like electromagnetism and fluid dynamics.

Vector Calculus

Vector calculus is the field of mathematics that deals with vector fields and operations on these fields. This branch is essential for fields such as physics and engineering, where it helps in modeling various phenomena.
Two primary operations in vector calculus are differentiation and integration of vector fields, which involve concepts such as gradients, divergence, and curl. Particularly in the given problem, the curl is crucial.

• The curl operation provides a vector that describes the rotation of a field at a certain point, which is pivotal in Stokes' Theorem.
• In our solution, we specifically obtained the curl of the vector field \(\mathbf{F}=(2z+x)\mathbf{i}+(y-z)\mathbf{j}+(x+y)\mathbf{k}\) and found it to be \(-2\mathbf{i} + 1\mathbf{j} + 1\mathbf{k}\).

Understanding how the curl affects the field at each point on a surface is crucial in applications that include electromagnetic fields and fluid movement. This domain reinforces the foundational knowledge necessary for solving problems involving force fields and energy distributions.

Parameterization of Surfaces

Parameterization is a method used to express surfaces with variables known as parameters. By parameterizing a surface, we essentially describe every point on it in terms of two other variables. This is particularly useful in integrating over surfaces, where it helps convert the region of integration into a more convenient form.
When dealing with more complex surfaces like those in vector calculus problems, parameterization allows us to streamline calculations.

• For the triangular surface in our problem, parameterization was done using parameters \(u\) and \(v\) with the equation \(\mathbf{r}(u,v) = u\mathbf{i} + v\mathbf{j} + (1-u-v)\mathbf{k}\).
• Limits for \(u\) and \(v\) were set to ensure coverage of the entire triangular region without exceedance.

This transformation is an integral part of applying Stokes’ Theorem, as it simplifies the multi-dimensional space into a workable calculation format. By understanding parameterization, you'll be well-equipped to handle complex integrations over surfaces and tackle a broad array of problems in mathematics and physics.