Question

Use Stokes' Theorem to evaluate the integral $$ \int_{C}-y^{3} d x+x^{3} d y-z^{3} d z $$ for \(C\) the (positively oriented) curve of intersection between the cylinder \(x^{2}+y^{2}=1\) and the plane \(x+y+z=1\)

Short answer

-0.3

Step-by-step solution

Find the Vector Field

The vector field \(\mathbf{F}\) can be written as: \[\mathbf{F} = -y^{3} \mathbf{i} + x^{3} \mathbf{j} - z^{3} \mathbf{k}\] By weighting the differential elements \(dx\), \(dy\), and \(dz\) by the physical quantities \(-y^3\), \(x^3\), and \(-z^3\) respectively.

Curl of the Vector Field

The curl of the field \(\mathbf{F}\) is calculated using \(\nabla \times \mathbf{F}\). The curl results to: \[\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ -y^{3} & x^{3} & -z^{3} \end{vmatrix} = (3z^2\mathbf{i} - 0\mathbf{j} - 3x^{2}\mathbf{k}) = 3z^{2}\mathbf{i} - 3x^{2}\mathbf{k}\]

Application of Stokes' Theorem

Integrate the flux of the curl over a disk \(D\) that lies in the plane \(z = 1 - x - y\), and is enclosed by the given curve \(C\): \[\int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_D (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\] where the surface \(S\) is positively oriented towards the upward direction \(n = \mathbf{k}\), the disk \(D\) has polar coordinates \(x = r\cos(\theta)\), \(y = r\sin(\theta)\), and \(z = 1 - r\). So our double integral simplifies into: \[\iint_D (\nabla \times \mathbf{F}) \cdot d\mathbf{S} =\iint_D [3r^2(1-r)\mathbf{i} - 3r^2\cos^2(\theta)\mathbf{k} ] \cdot \mathbf{k} \, rdrd\theta = -3 \int^{2\pi}_{0} \int^1_0 r^4 \cos^2(\theta) \, dr d\theta\]

Solve the Integral

Since the integrand does not depend on \(r\), we get \[-3 \int^{2\pi}_{0} \cos^2(\theta) d\theta \int^1_0 r^4 \, dr = -3 \cdot \frac{1}{2} \cdot \frac{1}{5} = -3/10 \] Note here that the integral of \(r^4\) dr from 0 to 1 is \(1/5\), and the integral of \(\cos^2(\theta)\) d\theta from 0 to \(2\pi\) is \(\pi\), hence you get the factor of \(\pi/2\).

Key concepts

Vector Field

A vector field assigns a vector to every point in a specific region of space. Picture a field of arrows where each arrow represents a vector in the field. In our example, the vector field \( \mathbf{F} \) is given by:\[ \mathbf{F} = -y^{3} \mathbf{i} + x^{3} \mathbf{j} - z^{3} \mathbf{k} \]Here, each component of the vector corresponds to one dimension in space, aligning with the coordinates \( x, y, \) and \( z \). The vector component \(-y^3\mathbf{i}\) acts along the x-direction, \(x^3\mathbf{j}\) along the y-direction, and \(-z^3\mathbf{k}\) along the z-direction.When you deal with vector fields in calculus, you explore how these vectors change or flow in space. Vector fields are core to understanding concepts like fluid flow, electromagnetic fields, and heat distribution.

Curl of a Vector Field

The curl of a vector field gives us an idea of the rotation or swirling strength of the field at any point. It tells us about the tendency to rotate about an axis. To find the curl \( abla \times \mathbf{F} \), we use a determinant involving partial derivatives:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \-y^{3} & x^{3} & -z^{3} \end{vmatrix} = 3z^{2}\mathbf{i} - 3x^{2}\mathbf{k} \]The result of this operation is a new vector field that points in the direction of the axis of rotation, with a magnitude that tells us how strongly the points in the field are swirling around that axis.

Surface Integral

A surface integral is a method of integrating a scalar or a vector field over a surface. It’s like laying a net over the surface and summing up the contributions from each piece of the net. The surface integral is crucial in applying Stokes' Theorem.In our context, we have a surface integral:\[ \iint_D (abla \times \mathbf{F}) \cdot d\mathbf{S} \]Here, \(d\mathbf{S}\) is the vector element of surface area, and the integral is over the region \(D\), a disk in our specific example. The components of \(abla \times \mathbf{F} \) are used in this integral to determine the net flow across the surface, contributing to the vortex or swirling patterns over the region.

Cylindrical Coordinates

To describe surfaces, curves, or bodies in three dimensions, it's often helpful to use cylindrical coordinates, especially when dealing with symmetrical shapes like cylinders. Cylindrical coordinates use a mixture of regular Cartesian and circular coordinates with \(r, \theta,\) and \(z\).- \(r\) is the radial distance from the axis of symmetry (usually the z-axis).- \(\theta\) is the angular coordinate, representing the rotation about the z-axis.- \(z\) is the height along the z-axis.In our problem, transforming the surface integral into cylindrical coordinates simplifies integration, as the cylinder’s equation \(x^2 + y^2 = 1\) directly converts to \(r = 1\), and \(z = 1 - x - y\) can also be adapted accordingly. This transformation reflects the adaptability and utility of cylindrical coordinates in evaluating integrals more conveniently.