Problem 364 - Calculus Volume 3

Question

Let \(S\) be paraboloid \(z=a\left(1-x^{2}-y^{2}\right),\) for \(z \geq 0, \quad\) where \(\quad a>0\) is a real number. Let \(\mathbf{F}=\langle x-y, y+z, z-x\rangle\). For what value(s) of \(a\) (if any) does \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) have its maximum value? For the following application exercises, the goal is to evaluate \(\quad A=\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S, \quad\) where \(\mathbf{F}=\langle x z,-x z, x y\rangle\) and \(S\) is the upper half of ellipsoid \(x^{2}+y^{2}+8 z^{2}=1,\) where \(z \geq 0\)

Step-by-Step Solution

Answer

The integral \( A \) is zero for any \( a > 0 \).

Step 1: Understand the problem

The problem requires determining the maximum value of the surface integral \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \). We have a vector field \( \mathbf{F} \) and a paraboloid surface \( S \). We need to explore the divergence theorem to convert the surface integral to a volume integral and find the value of \( a \) that maximizes it.

Step 2: Define the vector field \( \mathbf{F} \)

Given \( \mathbf{F} = \langle xz, -xz, xy \rangle \). This is the vector field over the paraboloid surface \( S \).

Step 3: Parameterize the surface \( S \)

The paraboloid is given by \( z = a(1-x^2-y^2) \) with \( z \geq 0 \). It can be parameterized as \( x = r\cos \theta \), \( y = r\sin \theta \), and \( z = a(1-r^2) \), where \( 0 \leq r \leq 1 \) and \( 0 \leq \theta < 2\pi \).

Step 4: Compute the curl \( \nabla \times \mathbf{F} \)

The curl of \( \mathbf{F} = \langle xz, -xz, xy \rangle \) is computed using \( abla \times \mathbf{F} = \left( \frac{\partial}{\partial y}(xy) - \frac{\partial}{\partial z}(-xz), \frac{\partial}{\partial z}(xz) - \frac{\partial}{\partial x}(xy), \frac{\partial}{\partial x}(-xz) - \frac{\partial}{\partial y}(xz) \right) \). Calculating these partial derivatives we have: \( abla \times \mathbf{F} = \langle y, z, -z - x \rangle \).

Step 5: Convert surface integral to volume integral using the Divergence Theorem

The Divergence Theorem states: \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \iiint_{V} abla \cdot (abla \times \mathbf{F}) \, dV \). Since the divergence of the curl of a vector field is always zero, \( abla \cdot (abla \times \mathbf{F}) = 0 \), this integral evaluates to zero over any volume \( V \). Thus, \( A = 0 \) irrespective of any parameter \( a \).

Step 6: Conclusion

Since the volume integral evaluates to zero regardless of the choice of \( a \), the surface integral will also be zero for any \( a > 0 \). Hence, \( a \) doesn't affect the integral's value in terms of maximizing it.

Key Concepts

Surface Integral

When dealing with a surface integral, we evaluate how much a vector field interacts with a surface. In simpler terms, it's like gauging the flow of a field across a surface.

For a vector field \( \mathbf{F} \), the surface integral over a surface \( S \) is given by \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS \), where \( \mathbf{n} \) is the normal vector to the surface.

This normal vector is crucial as it determines the direction in which the surface 'stands'. Think of \( \mathbf{F} \cdot \mathbf{n} \) as a measurement of the alignment between the vector field and this surface orientation.

• If \( \mathbf{F} \cdot \mathbf{n} \) is positive, the field tends to push outwards on the surface.
• If it's negative, the field pushes the surface inwards.
• If it's zero, there's no net flow across the surface.

Understanding surface integrals can be like watching how wind flows over a mountain or water streams in a river, which are both kinds of fields over surfaces.

Vector Field

A vector field is like a map of arrows spread across a region of space. Each arrow has both a magnitude and a direction, telling us how fast and where something like fluid flow or magnetic force moves. In mathematical terms, a vector field \( \mathbf{F} \) in three-dimensional space can be written in the form \( \mathbf{F} = \langle P(x,y,z), Q(x,y,z), R(x,y,z) \rangle \).

This means for each point in space (given by the coordinates \( x, y, z \)), the vector \( \mathbf{F} \) provides a distinct direction and intensity through its components \( P, Q, \) and \( R \).

Think of the vector field as the wind in the atmosphere, each arrow representing the wind's direction and speed at a specific location.

• Real-world phenomena, including gravitational, electric, and magnetic fields, can be modeled using vector fields.
• Vector fields are essential tools in engineering, physics, and meteorology.

Vector fields can become sophisticated, involving varying directions and strengths across a region, making them intriguing yet complex to study.

Paraboloid Surface

A Paraboloid Surface is a three-dimensional shape that looks like an infinite parabola spun around its axis. This surface is defined mathematically by a quadratic equation. Our specific paraboloid is represented by \( z = a(1-x^2-y^2) \), where \( z \geq 0 \).

This surface appears like an upward-facing bowl when graphed, stretching infinitely with a wide open mouth.

• Its shape is determined by the constant \( a \), which adjusts how steep or shallow the paraboloid appears.
• Paraboloid surfaces are ubiquitous in antenna designs and reflectors, thanks to their unique geometric properties.

The beauty of paraboloids lies in their symmetry and elegance, offering both mathematical intrigue and practical application.

Curl of a Vector Field

The curl of a vector field measures the rotation or 'twisting' at a point in the field. It's like observing the swirling movement of the clouds around a hurricane. Mathematically, for a vector field \( \mathbf{F} = \langle P, Q, R \rangle \), the curl is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \).

The resulting vector points in the direction of the axis of rotation and its magnitude shows how strong the rotation is at that point.

• If the curl is zero, the field is irrotational, indicating no local spinning.
• If not zero, the vector field exhibits a local rotational effect. Curl operations are crucial in fields such as fluid dynamics and electromagnetism.

In the context of the given exercise, calculating the curl helps determine how the vector field twists over the given paraboloid surface, ultimately aiding in evaluating the surface integral.