Problem 9 - Thomas Calculus

Question

Let \(S\) be the cylinder \(x^{2}+y^{2}=a^{2}, 0 \leq z \leq h,\) together with its top, \(x^{2}+y^{2} \leq a^{2}, z=h .\) Let \(\mathbf{F}=-y \mathbf{i}+x \mathbf{j}+x^{2} \mathbf{k} .\) Use Stokes' Theorem to find the flux of \(\nabla \times \mathbf{F}\) outward through \(S\) .

Step-by-Step Solution

Answer

The outward flux of \( \nabla \times \mathbf{F} \) through \( S \) is \( 2\pi a^2 \).

Step 1: Identify the Surface S

The surface \( S \) consists of a lateral cylindrical surface \( x^2 + y^2 = a^2 \) with \( 0 \leq z \leq h \) and the top cap defined by \( x^2 + y^2 \leq a^2, z = h \). We will use Stokes' Theorem to find the desired flux.

Step 2: Define Stokes' Theorem

Stokes' Theorem states that the flux of the curl of \( \mathbf{F} \) through surface \( S \) is equal to the line integral of \( \mathbf{F} \) along the boundary of \( S \). Mathematically, it is expressed as: \[ \int_S (abla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r} \] where \( C \) is the boundary of \( S \).

Step 3: Find the Curl of F

The vector field \( \mathbf{F} \) is given as \( \mathbf{F} = -y\mathbf{i} + x\mathbf{j} + x^2\mathbf{k} \). To find \( abla \times \mathbf{F} \), compute the determinant:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ -y & x & x^2 \end{vmatrix} \]Calculating, we find: \( abla \times \mathbf{F} = (0)\mathbf{i} + (0)\mathbf{j} + (2)\mathbf{k} = 2\mathbf{k} \).

Step 4: Calculate Flux Through Top Surface

For the top surface \( x^2 + y^2 \leq a^2, z = h \), the normal vector is \( \mathbf{k} \) and the surface element is \( dS = dx \ dy \ \mathbf{k} \). The flux through the top is:\[ \int_{S_{top}} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \int_{x^2 + y^2 \leq a^2} 2 \ dx \ dy = 2 \cdot \text{Area of circle} \]The area of the circle is \( \pi a^2 \), thus the flux is: \( 2\pi a^2 \).

Step 5: Consider the Lateral Surface

The flux through the cylindrical (lateral) surface is zero because the normal vector is perpendicular to the curl of \( \mathbf{F} \), which points in the \( z \)-direction. Thus, \( (abla \times \mathbf{F}) \cdot d\mathbf{S} = 0 \) for this surface.

Step 6: Combine Results

The total flux through the closed surface \( S \) is the flux through the top plus the flux through the lateral surface. Since the lateral flux is zero:\[ \int_S (abla \times \mathbf{F}) \cdot d\mathbf{S} = 2\pi a^2 \ + 0 \ = 2\pi a^2 \]

Key Concepts

Flux

Flux is a concept in vector calculus that describes the quantity of a field that passes through a given surface. In simple terms, you can think of flux as the flow of energy or particles through a surface. When dealing with fields like electric or magnetic fields, flux helps us quantify how much of that field is "flowing" through a particular area.

To calculate the flux, you integrate the field's vector field over the area, which involves considering both the field's strength and the angle at which it intersects the surface. The formula can be expressed as:

\[ \Phi = \int_S \mathbf{F} \cdot d\mathbf{S} \]

where \( \Phi \) is the flux, \( \mathbf{F} \) is the vector field, and \( d\mathbf{S} \) is the differential area vector of the surface \( S \).

This operation considers:

• The orientation of the surface: normal vectors point outward perpendicularly from the surface.
• The vector field's intensity: how strong the field is.

If the vector field is tangential to the surface, no flux passes through; if the field is perpendicular, it maximizes flux flow.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and their operations. It extends the methods of calculus to handle functions, which have multiple variables and directions.

Key concepts in vector calculus include:

• Gradients: These show the rate of change of a scalar field, like temperature, at a point in space.
• Divergence: This measures how much a field "spreads out" from a point. For instance, how air spreads out from a vent.
• Curl: A measure of how much a field "rotates" or circulates around a point.

Vector calculus is essential in fields such as electromagnetism, fluid dynamics, and physics in general, as it helps describe the physical phenomena involving vector fields.

In the problem involving Stokes' Theorem, vector calculus plays a crucial role in converting surface integrals to line integrals, making complex calculations often more manageable.

Curl

The curl of a vector field represents its rotational tendency or how much it "twirls" around a point. In simple terms, if you imagine the vector field as a flowing fluid, the curl helps you understand how much and the direction it turns or rotates around a point.

When computing the curl of a vector field \( \mathbf{F} \), you use the formula:

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

In our problem, the vector field is \( \mathbf{F} = -y\mathbf{i} + x\mathbf{j} + x^2\mathbf{k} \). Its curl is calculated using the determinant of a 3x3 matrix composed of unit vectors, partial derivatives, and the components of \( \mathbf{F} \), resulting in \( 2\mathbf{k} \). This indicates the direction and magnitude of rotation. The curl plays a vital role in applying Stokes' Theorem, which links the surface flux of the curl to a line integral over the boundary.

Cylinder Surface

A cylinder surface in geometry is a three-dimensional object with circular or elliptical base(s) and a set height. When you slice a cylinder vertically, the cross-sections are all identical circles.

In this problem, we have a combination of surfaces: the lateral cylindrical part and the top cap. The equation for the lateral surface of the cylinder is \( x^2 + y^2 = a^2 \) where \( 0 \leq z \leq h \), representing a classic cylindrical shape. The top cap is described by \( x^2 + y^2 \leq a^2, z = h \).

For vector calculus problems like Stokes' Theorem, understanding the cylinder surface is critical. Theneral:

• The lateral surface contributes zero flux since its normal vector is perpendicular to the curl of \( \mathbf{F} \).
• The top cap requires considering the curl's alignment with the surface normal.

Thus, correctly identifying and working with the cylindrical surface features is key to solving the problem and applying Stokes' Theorem effectively.