Question

In Exercises \(13-16,\) a closed surface \(S\) and a vector field \(\vec{F}\) are given. Find the outward flux of \(\vec{F}\) over \(S\) either through direct computation or through the Divergence Theorem. \(\mathcal{S}\) is the surface formed by the intersections of the cylinder \(z=1-x^{2}\) and the planes \(y=-2, y=2\) and \(z=0 ;\) \(\vec{F}=\left\langle 0, y^{3}, 0\right\rangle\)

Short answer

The flux is zero due to symmetric cancellations over y.

Step-by-step solution

Understand the surface

We need to understand the surface through which we will evaluate the flux. It is formed by the intersections of three surfaces: the cylinder described by the equation \(z = 1 - x^2\), and the planes \(y = -2\), \(y = 2\), and \(z = 0\). This forms a closed, bounded surface in three-dimensional space.

Analyze the vector field

The vector field is given by \(\vec{F} = \langle 0, y^3, 0 \rangle\). Notice that \(\vec{F}\) has components only in the y-direction, meaning only surfaces with a normal component in the y-direction will contribute to the flux.

Apply the Divergence Theorem

The Divergence Theorem states that the flux of \(\vec{F}\) across a closed surface \(S\) is equal to the triple integral of \(abla \cdot \vec{F}\) over the volume \(V\) enclosed by \(S\), \ \[\iint_S \vec{F} \cdot d\vec{S} = \iiint_V (abla \cdot \vec{F}) \, dV.\] First, compute \(abla \cdot \vec{F}\): \(abla \cdot \vec{F} = \frac{\partial}{\partial x}(0) + \frac{\partial}{\partial y}(y^3) + \frac{\partial}{\partial z}(0) = 3y^2.\)

Set up the volume integral

Now, integrate \(3y^2\) over the volume \(V\) defined by the surface. The limits for \(y\) are from \(-2\) to \(2\), for \(x\) we note the boundary from \(-\sqrt{1-z}\) to \(\sqrt{1-z}\) due to the cylinder surface, and for \(z\) from \(0\) to \(1-x^2\). Evaluate the integral: \[\iiint_V 3y^2 \, dV = \int_{-2}^{2}\int_{-\sqrt{1-z}}^{\sqrt{1-z}}\int_{0}^{1-x^2} 3y^2 \, dz \, dx \, dy.\]

Compute the triple integral

Now, perform the actual integration. Start with the innermost integral with respect to \(z\), since \(y\) is constant:\[\int_{0}^{1-x^2} 3y^2 \, dz = 3y^2(1-x^2).\]Proceed with the integration over \(x\):\[\int_{-\sqrt{1-z}}^{\sqrt{1-z}} 3y^2 (1-x^2) \, dx.\]Continue with integrating with respect to \(x\) and then \(y\) to complete the triple integral.

Integrate over x and y

Perform the integrations:- Over \(x\):\[\int_{-\sqrt{1-z}}^{\sqrt{1-z}} 3y^2 (1-x^2) \, dx\] yields a constant factor with respect to \(y\) after solving the quadratic integral.- Finally, compute the integral over \(y\): \[\int_{-2}^{2} \ldots \, dy\].

Key concepts

Vector Field

In mathematical physics, a vector field assigns a vector to every point in space. In our example exercise, the vector field is denoted by \( \vec{F} = \langle 0, y^3, 0 \rangle \). This means that at any point \((x, y, z)\), the vector field has only a component in the y-direction. This implies that the movement or force described by this vector field acts purely in the y-direction across all points.

When dealing with such vector fields, it's crucial to understand how they interact with surfaces that lie in the field. Specifically, when calculating quantities like flux, which measures how much of the vector field passes through a surface, we focus on how the vector field aligns with the normal vectors of the surface.

• The given vector field has a y-component, indicating it will significantly affect surfaces with normals in the y-direction.
• Surfaces with normals that strongly align with the y-direction will contribute more to the flux calculation.

Understanding the directionality of the vector field is an essential step before applying further mathematical theorems or techniques, like the Divergence Theorem.

Flux Calculation

Flux represents the amount of vector field passing through a surface. For a given vector field \(\vec{F}\), the flux across a closed surface \(S\) requires evaluating the surface integral \[\iint_S \vec{F} \cdot d\vec{S}.\]

This integral is calculated over the surface, taking into account how the vector field interacts with each differential area element of the surface. For simplicity, this integration can often utilize additional mathematical tools like the Divergence Theorem.

This theorem facilitates flux calculation by transforming a complex surface integral into a generally more manageable volume integral using the divergence of the vector field.

• The Divergence Theorem states that the flux of \(\vec{F}\) through \(S\) is equal to the triple integral of the divergence of \(\vec{F}\) over the volume \(V\) enclosed by \(S\).
• Finding the divergence simplifies the task by using properties of volume rather than directly dealing with the surface.

Triple Integral

The triple integral is an extension of the integral calculus to three dimensions and is crucial in evaluating quantities like flux over a volume \(V\). In our problem setup, the triple integral involves\[\iiint_V 3y^2 \, dV,\]
moving through the iterated integration of variables, namely \(x\), \(y\), and \(z\).

Solving a triple integral involves critical steps including:

• Setting proper limits for each integral based on the geometry of the volume.
• Choosing the order of integration which can simplify the evaluation process depending on symmetry and functional dependencies.
• Moving consistently from innermost to outermost integral while applying the fundamental theorem of calculus.

In our case, integrating the given function \(3y^2\) over the defined boundaries allows us to compute the total flux through transformations applied by the Divergence Theorem.

Cylindrical Coordinates

Cylindrical coordinates provide an alternative to Cartesian coordinates for describing geometries that contain quantities displaying radial symmetry, such as our cylindrical surface \(z = 1 - x^2\).

These coordinates are advantageous when dealing with problems involving symmetrical cylinders because they simplify the mathematical description of the system:

• The coordinates involve \(r\) (radius), \(\theta\) (angle), and \(z\) (height), making them ideal for problems with rotational symmetry.
• Conversion between Cartesian and cylindrical coordinates can streamline computation, particularly when setting limits and integrals.

In our example, using cylindrical coordinates for the given geometry would require expressing the surface and vector field in these terms, potentially easing the integration process. Though the original problem mainly describes \((x,y,z)\)-space, understanding this transformation is beneficial for similar problems.