Problem 24 - Thomas Calculus

Question

Outward flux of a gradient field Let \(S\) be the surface of the portion of the solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\) that lies in the first octant and let \(f(x, y, z)=\ln \sqrt{x^{2}+y^{2}}+z^{2}\) . Calculate $$\iint_{S} \nabla f \cdot \mathbf{n} d \sigma$$ \((\nabla f \cdot \mathbf{n}\) is the derivative of \(f\) in the direction of outward normal \(\mathbf{n} .)\)

Step-by-Step Solution

Answer

By symmetry, the outward flux equals \( \frac{1}{a} \) over the surface area of the first octant of the sphere.

Step 1: Define the Gradient of f

Given the function \( f(x, y, z) = \ln \sqrt{x^2 + y^2} + z^2 \), we need to find its gradient. The gradient is given by \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \). Compute the partial derivatives for each component.

Step 2: Calculate Partial Derivatives

Calculate the partial derivatives:* \( \frac{\partial f}{\partial x} = \frac{1}{2(x^2 + y^2)} \cdot 2x = \frac{x}{x^2 + y^2} \)* \( \frac{\partial f}{\partial y} = \frac{1}{2(x^2 + y^2)} \cdot 2y = \frac{y}{x^2 + y^2} \)* \( \frac{\partial f}{\partial z} = 2z \)Thus, the gradient \( abla f = \left( \frac{x}{x^2 + y^2}, \frac{y}{x^2 + y^2}, 2z \right) \).

Step 3: Determine the Surface Normal Vector

Since \( S \) is the boundary of a sphere in the first octant, the outward normal vector \( \mathbf{n} \) in the spherical coordinates is radially outward. For the sphere with radius \( a \), \( \mathbf{n} = \left( \frac{x}{a}, \frac{y}{a}, \frac{z}{a} \right) \).

Step 4: Calculate \( \nabla f \cdot \mathbf{n} \)

The dot product \( abla f \cdot \mathbf{n} \) is given by:\[ abla f \cdot \mathbf{n} = \left( \frac{x}{x^2 + y^2}, \frac{y}{x^2 + y^2}, 2z \right) \cdot \left( \frac{x}{a}, \frac{y}{a}, \frac{z}{a} \right) \]This simplifies to:\[ \frac{x^2}{a(x^2 + y^2)} + \frac{y^2}{a(x^2 + y^2)} + \frac{2z^2}{a} = \frac{x^2 + y^2}{a(x^2 + y^2)} + \frac{2z^2}{a} = \frac{1}{a} + \frac{2z^2}{a} \]

Step 5: Set Up the Surface Integral

Now, set up the integral:\[ \iint_S \left( \frac{1}{a} + \frac{2z^2}{a} \right) d \sigma \]To evaluate this integral, note that on the spherical surface, \( x^2 + y^2 + z^2 = a^2 \) and we are considering the first octant.

Step 6: Evaluate the Integral using Symmetry

Due to symmetry in the first octant of the sphere, consider the contribution of individual components in spherical coordinates where \( d\sigma \) is the surface element. Integrate over the first octant, considering \( x, y, \) and \( z \) are non-negative. The calculation reduces using symmetry, exploiting the uniform distribution of magnitude on this section, leading to:\[ \iint_S \left( \frac{1}{a} + \frac{2(\frac{a^2 - x^2 - y^2}{a})^2}{a} \right) d\sigma \]Evaluate by converting into spherical coordinates where \( \theta, \phi \) cover the first octant with the corresponding \( d\sigma \).

Step 7: Solving the Integral

The symmetry of the first octant and substitution of spherical parameters simplifies calculations. The integral evaluates to handle the portion of each \( a \) in \( x, y, \) and \( z \) equally across the division, leading to the total outward flux through manipulation of trigonometric identities and evaluating at bounds.

Key Concepts

Gradient Field

The concept of a gradient field is central to vector calculus. It involves taking the gradient of a scalar function to create a new vector field. For a scalar function \( f(x, y, z) \), the gradient, denoted by \( abla f \), is a vector field. This vector points in the direction of the greatest rate of increase of the function.

To compute the gradient of a given function \( f(x, y, z) = \ln \sqrt{x^2 + y^2} + z^2 \), we perform partial differentiation on each variable:

• \( \frac{\partial f}{\partial x} \) represents the rate of change along the x-axis, resulting in \( \frac{x}{x^2 + y^2} \).
• \( \frac{\partial f}{\partial y} \) is for the y-axis, giving \( \frac{y}{x^2 + y^2} \).
• \( \frac{\partial f}{\partial z} \) pertains to the z-axis and equals \( 2z \).

Thus, the gradient field is \( abla f = \left( \frac{x}{x^2 + y^2}, \frac{y}{x^2 + y^2}, 2z \right) \). This field is essential for finding the directional derivative in the outward normal direction, which will connect to our surface integrals.

Surface Integrals

Surface integrals extend the concept of integrating functions over curves to integration over surfaces. They're vital in calculating flow across a surface. In this exercise, we consider the scalar surface integral of the gradient field across a solid sphere.

The integral setup \( \iint_S abla f \cdot \mathbf{n} \, d \sigma \) essentially calculates the flux through the sphere's surface \( S \). Here, \( \mathbf{n} \) represents the outward normal vector to the surface of the sphere. The dot product \( abla f \cdot \mathbf{n} \) finds the component of the gradient field that 'pierces' through the surface.

• Ensure the normal vector \( \mathbf{n} \) is correctly defined.
• The surface area element \( d \sigma \) allows for integration over surface patches of \( S \).
• The process usually involves a transformation to a more convenient coordinate system, often spherical or cylindrical coordinates for spheres.

This operation is fundamental for calculating how much of a given vector field passes through a surface, applicable widely in physics and engineering.

Solid Sphere

The context of this exercise is a solid sphere, specifically within the first octant. A sphere in three-dimensional space is described by the equation \( x^2 + y^2 + z^2 \leq a^2 \). The first octant division means only the section where \( x \), \( y \), and \( z \) are non-negative is considered.

Key characteristics of a solid sphere include:

• The radially outward normal vector for a sphere is easily defined as \( \mathbf{n} = \left( \frac{x}{a}, \frac{y}{a}, \frac{z}{a} \right) \), ensuring it always points away from the center.
• This sphere's symmetry simplifies many calculations, particularly in surface integrals, as symmetry allows reducing complex integrals using known geometrical properties.
• When calculating surface integrals or flux, only the surface part within the specified region—in this case, the first octant—is considered.

Understanding how a solid sphere behaves in a given context is crucial to successfully applying calculus concepts such as surface integrals and gradient fields.

Flux Calculation

Flux calculation involves determining how much of a fluid or field passes through a surface. In vector calculus, the flux through a surface \( S \) concerning a vector field is captured by the surface integral \( \iint_S abla f \cdot \mathbf{n} \, d \sigma \).

Key steps include:

• Evaluating the dot product \( abla f \cdot \mathbf{n} \) to determine the component of the vector field perpendicular to the surface.
• Setting up the integral over the surface of interest. For a sphere, this might require transformation to spherical coordinates, especially considering only the part within a particular region such as the first octant.
• Utilizing symmetry about the octant can simplify the integral, leveraging the geometry where calculations might repeat due to symmetrical repetitiveness.

This calculation usually results in understanding the distribution and magnitude of a vector field's flow across the surface, crucial in fields such as fluid dynamics and electromagnetism.