Question

Use either Stokes' theorem or the divergence theorem to evaluate each of the following integrals in the easiest possible way. \(\iint \operatorname{curl}\left(x^{2} y \mathbf{i}-x z \mathbf{k}\right) \cdot \mathbf{n} d \sigma\) over the closed surface of the ellipsoid $$\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1$$ Warning: Stokes' theorem applies only to an open surface. Hints: Could you cut the given surface into two halves? Also see (d) in the table of vector identities (page 339).

Short answer

The integral evaluates to zero.

Step-by-step solution

Identify the appropriate theorem

Given that Stokes' theorem applies to open surfaces and the problem involves a closed surface, use the divergence theorem. The divergence theorem states that \[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{V} abla \cdot \mathbf{F} \, dV \] where \( \mathbf{F} \) is a vector field and \( abla \cdot \mathbf{F} \) is the divergence of \( \mathbf{F} \).

Find the vector field and its divergence

The given vector field is \( \mathbf{F} = x^2 y \mathbf{i} - x z \mathbf{k} \). Find the divergence of \( \mathbf{F} \): \( abla \cdot \mathbf{F} = \frac{\partial (x^2 y)}{\partial x} + \frac{\partial (0)}{\partial y} + \frac{\partial (-xz)}{\partial z} \). Calculating the partial derivatives: \( \frac{\partial (x^2 y)}{\partial x} = 2xy \) \( \frac{\partial (0)}{\partial y} = 0 \) \( \frac{\partial (-xz)}{\partial z} = -x \) Therefore, \( abla \cdot \mathbf{F} = 2xy - x \).

Set up the integral over the volume

Using the divergence theorem, convert the surface integral to a volume integral: \( \iint_{S} \operatorname{curl}(x^2 y \mathbf{i} - x z \mathbf{k}) \cdot \mathbf{n} \, d \sigma = \iiint_{V} (2xy - x) \, dV \) Here, the volume \( V \) is the volume of the ellipsoid given by \( \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1 \).

Transform the ellipsoid to a standard sphere using a change of variables

To simplify the calculation, transform the ellipsoid into a standard sphere using the following variable changes: \( u = \frac{x}{2}, \, v = \frac{y}{3}, \, w = \frac{z}{4} \) This transformation changes the ellipsoid equation to \( u^2 + v^2 + w^2 = 1 \) and the volume element to \( dV = 24 \, du \, dv \, dw \), taking into account the scaling factors.

Integrate over the new variables

Substitute the new variables into the integral: \( x = 2u, \, y = 3v \) Thus, \( 2xy - x = 12uv - 2u \). Transform the volume integral: \( \iiint_{V} (2xy - x) \, dV = 24 \iiint_{u^2 + v^2 + w^2 \leq 1} (12uv - 2u) \, du \, dv \, dw \).

Evaluate the integral over the unit sphere

Note that integration of linear terms (like \(2u\)) over the symmetric limits of the unit sphere results in zero because the sphere's volume is symmetric in \(u, v\) and \(w\). Therefore, \( \iiint_{u^2 + v^2 + w^2 \leq1} (12uv - 2u) \, du \, dv \, dw = 0 \).

Conclude the result

The integral evaluates to zero. Thus: \( \iint_{S} \operatorname{curl}(x^2 y \mathbf{i} - x z \mathbf{k}) \cdot \mathbf{n} \, d \sigma = \iiint_{V} (2xy - x) \, dV = 0 \).

Key concepts

Surface Integral

A surface integral is a core concept in vector calculus, representing the integral of a scalar field or vector field over a surface. Think of it as summing up the values of a field across every infinitesimal piece of a surface. When dealing with vector fields, like in our problem, the surface integral is expressed in terms of the field \( \mathbf{F} \cdot \mathbf{n} \ dS \), where \( \mathbf{F} \) is the vector field, \( \mathbf{n} \) is the unit normal vector to the surface, and \( \ dS \) is an infinitesimal surface element. This integral takes into account both the magnitude of the field and its direction relative to the surface. The Divergence Theorem helps convert a complex surface integral into a more manageable volume integral, simplifying computations significantly.

Vector Field

A vector field is a function that assigns a vector to every point in space. In our problem, the vector field is given by \( \mathbf{F} = x^2 y \mathbf{i} - x z \mathbf{k} \). We use vector fields to model physical fields such as gravitational, electric, and velocity fields. Each vector in this field represents both the direction and magnitude of the physical quantity at that point.
To apply the Divergence Theorem, we need the divergence of the vector field, symbolized by \( \ abla \cdot \mathbf{F} \). The divergence measures the net rate of flux expansion at a point: it tells us how much the vector field 'spreads out.' For our field, the divergence is \( 2xy - x \), obtained by taking partial derivatives with respect to x, y, and z.

Ellipsoid

An ellipsoid is a three-dimensional geometric shape, much like an elongated or squashed sphere. Parametrically, it is defined by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). Our problem deals with an ellipsoid specified by \( \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 1 \) which we transform into a unit sphere to simplify integration.
To do the transformation, we rescale the variables: \( x = 2u, y = 3v, z = 4w \). This converts the ellipsoid into a sphere \( u^2 + v^2 + w^2 = 1 \), making integration straightforward with the new volume element \( 24 \ du \ dv \ dw \). This step is crucial because integrating over a sphere with symmetric limits results in significant simplification, illustrated by the fact that terms like \( 2u \) over symmetric limits results in zero.