Problem 367 - Calculus Volume 3

Question

Take paraboloid \(z=x^{2}+y^{2},\) for \(0 \leq z \leq 4,\) and slice it with plane \(y=0\). Let \(S\) be the surface that remains for \(y \geq 0\), including the planar surface in the \(x z\) -plane. Let \(C\) be the semicircle and line segment that bounded the cap of \(S\) in plane \(z=4\) with counterclockwise orientation. Let \(\quad \mathbf{F}=\langle 2 z+y, 2 x+z, 2 y+x\rangle\). Evaluate \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\)

Step-by-Step Solution

Answer

The integral over the surface \(S\) is \(2\pi\).

Step 1: Understanding the Surface and Vector Field

The surface given is part of the paraboloid described by the equation \(z = x^2 + y^2\) for \(0 \leq z \leq 4\), further sliced by the plane \(y=0\), and we consider only the region where \(y \geq 0\). The cap at \(z=4\) lies in the plane with the equation \(x^2 + y^2 = 4\), set in the semicircle \(y \geq 0\). The vector field given is \(\mathbf{F} = \langle 2z + y, 2x + z, 2y + x \rangle\). We are tasked with evaluating \(\iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS\).

Step 2: Calculate the Curl of \(\mathbf{F}\)

First, we calculate the curl of \(\mathbf{F}\) using the formula \(abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\). Calculating each term: - \(\frac{\partial F_3}{\partial y} = 2\), \(\frac{\partial F_2}{\partial z} = 1\)- \(\frac{\partial F_1}{\partial z} = 2\), \(\frac{\partial F_3}{\partial x} = 1\)- \(\frac{\partial F_2}{\partial x} = 2\), \(\frac{\partial F_1}{\partial y} = 1\)Thus, \(abla \times \mathbf{F} = \langle 1, 1, 1 \rangle\).

Step 3: Parameterize the Surface \(S\)

The surface \(S\) consists of two parts: the paraboloid segment \(z = x^2 + y^2\) for \(0 \leq z \leq 4, y \geq 0\) and the flat semicircle cap at \(z = 4\). For the cap, we use polar coordinates with \(x = r \cos \theta, y = r \sin \theta, r = 2, 0 \leq \theta \leq \pi\).

Step 4: Calculate the Integral over the Cap

For the cap of the surface at \(z = 4\), the normal vector pointing outward is \(\mathbf{n} = \langle 0, 0, 1 \rangle\). The dot product \((abla \times \mathbf{F}) \cdot \mathbf{n} = 1 \cdot 1 = 1\). The surface area element \(dS = r \, dr \, d\theta\). Set up the integral: \[ \iint_{cap} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \int_{0}^{2} \int_{0}^{\pi} 1 \cdot r \, dr \, d\theta \]Evaluate the integral: \[ \int_{0}^{2} r \, dr = \left. \frac{r^2}{2} \right|_{0}^{2} = 2 \]\[ \int_{0}^{\pi} d\theta = \pi \]Thus, the integral over the cap is \(2\pi\).

Step 5: Stokes' Theorem Confirmation (Optional)

From Stokes' Theorem, the surface upward flux of \((abla \times \mathbf{F})\) through \(S\) should equal the line integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\) along \(C\), the boundary of \(S\). Here, ensuring the boundary of the surface \(S\) considered is a closed curve helps check the consistency of using Stokes' Theorem.

Key Concepts

Curl of a Vector Field

The concept of the "curl" in vector calculus is crucial for understanding how a vector field behaves when it rotates around a point or along a surface. **Curl**, symbolized as \( abla \times \mathbf{F} \), is a way to measure the rotational effect of the vector field \( \mathbf{F} \) within a space. Think of it like the twisting motion of a small paddle wheel placed in the field; it tells you how intensely the field is swirling around at anyone point.
To calculate the curl of a vector field, use the formula:
\[ abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \]
For the given vector field \( \mathbf{F} = \langle 2z + y, 2x + z, 2y + x \rangle \), the computed curl is \( \langle 1, 1, 1 \rangle \). This uniform result represents an equal and consistent swirling motion in all directions within the field across the surface.

Parametrizing Surfaces

Parametrization of surfaces is a neat way to describe complex shapes using simpler coordinate systems such as polar coordinates. This simplification is useful because it transforms complicated surfaces into more manageable forms.
In the given exercise, the paraboloid surface \( z = x^2 + y^2 \), within the region for \( 0 \leq z \leq 4 \), and its cap at \( z = 4 \), can be parameterized using polar coordinates. For instance, for the cap where \( x^2 + y^2 \leq 4 \) and \( y \geq 0 \), parameters can be:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( z = 4 \)
• \( 0 \leq r \leq 2 \) and \( 0 \leq \theta \leq \pi \)

This parameterization is beneficial as it allows you to describe the shape and surface elements more conveniently for integration.

Integration Over a Surface

When performing integration over a surface, an essential component is to evaluate how a vector field behaves relative to that surface. This process is called a **surface integral**. It's akin to summing up all the small influences exerted by a vector field across an entire surface.
The general surface integral to evaluate is \( \iint_{S} \mathbf{G} \cdot \mathbf{n} \, dS \), where \( \mathbf{G} \) is some vector field, \( \mathbf{n} \) is the normal vector to the surface, and \( dS \) is the differential area of the surface. In our exercise, we calculated:

• \( \mathbf{n} = \langle 0, 0, 1 \rangle \) for the cap at \( z = 4 \)
• The integral over the cap area involved polar parameters and was evaluated as \( 2\pi \)

This final value represents the sum of the field's effects over the entire cap surface.

Vector Calculus

Vector calculus is an advanced branch of mathematics that deals with vector fields and operations on them like differentiation and integration. It's instrumental in analyzing fields like electromagnetism and fluid dynamics, where quantities change across space.
In this exercise, vector calculus encompasses several steps:

• Calculating the **curl** of the vector field, \( abla \times \mathbf{F} \)
• Parametrizing the curved surfaces
• Integrating over these surfaces to derive required physical interpretations

Understanding vector calculus helps in grasping how multivariate signals interact, and how real-world applications, like fluid flow or electromagnetic fields, behave in complex environments.

Surface Integrals

A surface integral is a vital concept in mathematics and physics when you need to understand how a field influences a surface. Just like a line integral evaluates a function along a curve, a surface integral evaluates a function across a surface.
To compute a surface integral, express the surface in question using parameters (like polar coordinates), find the differential surface area \( dS \), and then perform \( \iint_{S} \) operations over the entire surface.
In this problem:

• For the cap \( z = 4 \), we calculated \( \iint (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \) with \( \mathbf{n} = \langle 0, 0, 1 \rangle \)
• The result of this integral gave us \( 2\pi \), illustrating how the vector field penetrates the surface.

Surface integrals provide us with a comprehensive overview, quantifying the complete effect of a field across a surface.