Problem 331 - Calculus Volume 3

Question

For the following exercises, without using Stokes' theorem, calculate directly both the flux of \(\operatorname{curl} \mathbf{F} \cdot \mathbf{N}\) over the given surface and the circulation integral around its boundary, assuming all boundaries are oriented clockwise as viewed from above. \(\mathbf{F}(x, y, z)=2 y \mathbf{i}-6 z \mathbf{j}+3 x \mathbf{k} ; \quad S\) is a portion of paraboloid \(z=4-x^{2}-y^{2}\) and is above the \(x y\) -plane.

Step-by-Step Solution

Answer

The flux through the surface is zero after integrating and evaluating the terms.

Step 1: Compute the Curl of F

The curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by: \[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]For \( \mathbf{F}(x, y, z)=2 y \mathbf{i}-6 z \mathbf{j}+3 x \mathbf{k} \), compute each component:- \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = \frac{\partial}{\partial y}(3x) - \frac{\partial}{\partial z}(-6z) = 0 + 6 = 6 \)- \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 0 - 3 = -3 \)- \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 - 2 = -2 \)Thus, \( abla \times \mathbf{F} = 6 \mathbf{i} - 3 \mathbf{j} - 2 \mathbf{k} \).

Step 2: Determine the Normal Vector of the Surface S

For the surface \( z = 4 - x^2 - y^2 \), consider the surface as given by \( r(x, y) = x \mathbf{i} + y \mathbf{j} + (4 - x^2 - y^2) \mathbf{k} \). The normal vector \( \mathbf{N} \) can be found via the cross product of the partial derivatives:\[ \mathbf{N} = r_x \times r_y \]Find:- \( r_x = \frac{\partial}{\partial x}(x, y, 4-x^2-y^2) = (1, 0, -2x) \)- \( r_y = \frac{\partial}{\partial y}(x, y, 4-x^2-y^2) = (0, 1, -2y) \)Compute the cross product:\[ \mathbf{N} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \1 & 0 & -2x \0 & 1 & -2y\end{vmatrix}= (2x \mathbf{i} + 2y \mathbf{j} + \mathbf{k}) \]

Step 3: Calculate the Flux of Curl F Over S

The flux through the surface is given by \( \iint_S (abla \times \mathbf{F}) \cdot \mathbf{N} \, dS \).Substitute \( abla \times \mathbf{F} = 6 \mathbf{i} - 3 \mathbf{j} - 2 \mathbf{k} \) and \( \mathbf{N} = 2x \mathbf{i} + 2y \mathbf{j} + \mathbf{k} \):\[ \iint_S (6 \mathbf{i} - 3 \mathbf{j} - 2 \mathbf{k}) \cdot (2x \mathbf{i} + 2y \mathbf{j} + \mathbf{k}) \, dS = \iint_S (12x - 6y - 2) \, dS \]Since the region is defined by \( z = 4 - x^2 - y^2 \), the region on the \( xy \)-plane is \( x^2 + y^2 \leq 4 \). Convert to polar coordinates with \( x = r\cos\theta \) and \( y = r\sin\theta \), where \( dS = r \, dr \, d\theta \):\[ \iint_R (12r\cos\theta - 6r\sin\theta - 2) r \, dr \, d\theta = \int_0^{2\pi} \int_0^2 (12r^2\cos\theta - 6r^2\sin\theta - 2r) \, dr \]}]{

Key Concepts

Flux Calculation

Flux calculation is an important concept in vector calculus that refers to the amount of a field that passes through a surface. In this context, we care about the flux of the curl of a vector field over a given surface. The expression is mathematically represented by \( \iint_S (abla \times \mathbf{F}) \cdot \mathbf{N} \, dS \), where \( abla \times \mathbf{F} \) represents the curl of the vector field \( \mathbf{F} \), \( \mathbf{N} \) is the normal vector to the surface \( S \), and \( dS \) represents a small area element on the surface.

To compute the flux, you take the dot product of the curl of the vector field with the normal vector and integrate over the whole surface. This process helps in understanding how a vector field behaves over a surface. For example, if the vector field represents fluid flow, the flux indicates how much fluid is passing through the surface, which can be instrumental in fields like electromagnetism and fluid dynamics.

Curl of a Vector Field

The curl of a vector field measures the rotation or swirling strength of the field at any given point. It's calculated using the cross product of the del operator (\( abla \)) with the vector field \( \mathbf{F} \). The formula is:

\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]

This operation helps us determine whether fluid or air in a space tends to rotate around any particular axis or if the field lines rotate within a three-dimensional space. Such properties are crucial when studying electromagnetism or fluid flow. For example, knowing the swirl within a magnetic field is vital for understanding how electromagnetic devices function.

Surface Integrals

Surface integrals are used to integrate over a surface in three-dimensional space. It extends the concept of a line integral to surfaces. When you calculate the surface integral of a vector field \( \mathbf{F} \), you are essentially summing up a vector quantity over a surface, much like you would sum scalar values in calculus.

A typical application is finding the flux of a field which, in simpler terms, measures total "flow" on a surface. To compute it, you must:

• Determine a parameterization of the surface.
• Calculate the normal vector \( \mathbf{N} \) of the surface.
• Perform the integral \( \iint_S \mathbf{F} \cdot \mathbf{N} \, dS \).

These techniques are often used in electromagnetic theory to compute magnetic or electric flux, aiding in designing circuits and electric devices.

Polar Coordinates

Polar coordinates provide a two-dimensional coordinate system where each point is determined by a distance from a reference point and an angle from a reference direction. This system is particularly useful in calculating surface integrals over circular regions.

When transforming from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\), you use:

• \( x = r \cos\theta \)
• \( y = r \sin\theta \)
• \( dS = r \, dr \, d\theta \) for area elements

These transformations simplify integrals when the region of interest is circular or symmetrical in nature. Opting for polar coordinates often reduces computational complexity by aligning the integration boundaries along constant radii and angles, making the calculations more intuitive and manageable.

Normal Vector

A normal vector is perpendicular to a given surface at any point and is crucial for calculating the surface integral. It essentially points outward relative to the surface, like how surface direction could extend from a balloon's exterior in all directions.

For complex surfaces, the normal vector can be obtained by taking the cross product of two vectors that are tangent to the surface. If your surface \( S \) is given by a parametric equation \( \mathbf{r}(x, y) \), then the normal vector \( \mathbf{N} \) is:

\[ \mathbf{N} = \mathbf{r}_x \times \mathbf{r}_y \]

This calculation ensures that the vector is orthogonal to every tangent line on the surface, allowing accurate computation of the flow across the surface by using it in conjunction with the vector field in a flux calculation. Understanding normal vectors is essential for fields such as computer graphics and engineering simulations.