Question

Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use. $$\begin{aligned} &\mathbf{F}=\left\langle x^{2} e^{y} \cos z,-4 x e^{y} \cos z, 2 x e^{y} \sin z\right\rangle ; S \text { is the boundary of }\\\ &\text { the ellipsoid } x^{2} / 4+y^{2}+z^{2}=1 \end{aligned}$$

Short answer

In this question, we found the outward flux of the given vector field $\mathbf{F} = \left\langle x^2 e^y \cos z, -4x e^y \cos z, 2x e^y \sin z \right\rangle$ across the surface S defined by the ellipsoid $\frac{x^2}{4} + y^2 + z^2 = 1$. By applying the Divergence theorem, we computed the divergence of the vector field and set up a triple integral using spherical coordinates to compute the flux. In the end, we calculated the flux to be equal to 0.

Step-by-step solution

Compute the factor multiplying with coordinates

The factor multiplies with coordinates x, y, and z are: $$f_x = x^2e^y \cos z$$ $$f_y = -4x e^y \cos z$$ $$f_z = 2x e^y \sin z$$

Compute the divergence of the vector field

The divergence of the vector field is the dot product of the gradient and the field, which can be computed as follows: $$\nabla \cdot \mathbf{F} = \frac{\partial f_x}{\partial x} + \frac{\partial f_y}{\partial y} + \frac{\partial f_z}{\partial z}$$ Calculating the derivatives, we get: $$\frac{\partial f_x}{\partial x} = 2xe^y\cos z$$ $$\frac{\partial f_y}{\partial y} = -4xe^y\cos z$$ $$\frac{\partial f_z}{\partial z} = 2xe^y\cos z$$ Now, we can find the divergence: $$\nabla \cdot \mathbf{F} = 2xe^y\cos z - 4xe^y\cos z + 2xe^y\cos z = 0$$

Convert the ellipsoid equation to a parametric form

Next, we need to convert the ellipsoid equation into a parametric form, using spherical coordinates. With this conversion, we will be able to set appropriate bounds for integration. Let us use the following substitutions: $$ x = 2\sin \phi \cos\theta$$ $$ y = \sin \phi \sin\theta$$ $$ z = \cos \phi$$ The corresponding Jacobian for this transformation is \(|J| = 2\sin^{2} \phi\) With these substitutions, the bounds for integration are given as follows: $$\phi \in \left[0,\pi\right]$$ $$\theta \in \left[0,2\pi\right]$$ $$\rho \in \left[0,1\right]$$

Set up the triple integral to compute the flux

Since the divergence is zero, the triple integral of the divergence will also be zero. Therefore, the outward flux through the surface would be: $$\text{Flux} = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{1} (\nabla \cdot \mathbf{F})\cdot |J|\, d\rho\, d\phi\, d\theta = 0 $$

Calculate the flux

So, the outward flux of the vector field across the given surface S is: $$\text{Flux} = 0$$

Key concepts

Divergence Theorem

The Divergence Theorem is a key concept in vector calculus. It's also known as Gauss's Theorem or Ostrogradsky's Theorem. The theorem offers a powerful way to transform a complicated surface integral into a more approachable volume integral by relating the flow of a vector field across a closed surface to the behavior of the field within the volume enclosed by the surface.

Mathematically, the Divergence Theorem is expressed as: \[\int_{S} \mathbf{F} \cdot d\mathbf{S} = \int_{V} abla \cdot \mathbf{F} \, dV\] where \( S \) is a closed surface, \( V \) is the volume enclosed by \( S \), \( \mathbf{F} \) the vector field, and \( abla \cdot \mathbf{F} \) represents the divergence of \( \mathbf{F} \).

This theorem is fundamental in physics and engineering as it allows for the conversion between the macroscopic supporting flux through a surface to the microscopic behavior inside a volume, simplifying problems involving complex surfaces.

Vector Fields

In calculus, a vector field assigns a vector to every point in space, conceptualizing physical phenomena like wind velocity through vector arrows pointing in the direction of the field, with their lengths corresponding to the field's magnitude.

In mathematical descriptions, a vector field in three-dimensional space is often expressed using components for each of the x, y, and z dimensions, as seen in \( \mathbf{F} = \langle f_x, f_y, f_z \rangle \), with each of these components possibly a function of x, y, and z.

The concept of divergence is intrinsically linked to vector fields, representing the rate at which "stuff" spreads out from a point. If you think of a vector field as depicting the flow of a fluid, areas of positive divergence represent areas where the fluid is spreading out, while negative divergence indicates areas where the fluid is converging.

Ellipsoid

An ellipsoid is an important geometric surface in mathematics, particularly within calculus and geometry, that generalizes the concept of an ellipse to three dimensions. Mathematically, an ellipsoid is often given by the equation: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\] where \( a \), \( b \), and \( c \) are the parameters defining the semi-axis lengths along the x, y, and z axes respectively.

In our discussion involving vector fields and the Divergence Theorem, the ellipsoid serves as a boundary surface over which we calculate the outward flux of the vector field.

Because of its rounded, smooth characteristics, the ellipsoid makes calculations using the Divergence Theorem more approachable, especially when converting to spherical coordinates, which adapt to the symmetry of the structure and simplify integration bounds, as shown in this problem's solution.