Question

Evaluate each integral in the simplest way possible. \(\iint \mathbf{V} \cdot \mathbf{n} d \sigma\) over the entire surface of the sphere \((x-2)^{2}+(y+3)^{2}+z^{2}=9,\) if $$\mathbf{V}=(3 x-y z) \mathbf{i}+\left(z^{2}-y^{2}\right) \mathbf{j}+\left(2 y z+x^{2}\right) \mathbf{k}$$

Short answer

The value of \( \iint \mathbf{V} \cdot \mathbf{n} \ d \sigma \) over the sphere is zero

Step-by-step solution

Understand the Exercise

We are given a vector field \(\textbf{V}\) and asked to find the surface integral of \(\textbf{V} \cdot \textbf{n} \ d \sigma\) over a sphere. To make this simpler, we can use the Divergence Theorem.

Set Up the Divergence Theorem

The Divergence Theorem states:\[\iint_S \mathbf{V} \cdot \mathbf{n} \ d \sigma = \iiint_V (\abla \cdot \mathbf{V}) \ dV\]Where \(S\) is the surface and \(V\) is the volume inside. We need to compute \(\abla \cdot \mathbf{V}\).

Calculate the Divergence of \( \mathbf{V} \)

Given \( \mathbf{V} = (3x - yz) \mathbf{i} + (z^2 - y^2) \mathbf{j} + (2yz + x^2) \mathbf{k}\), we find the divergence by applying \( \abla \cdot \mathbf{V} = \frac{\partial (3x - yz)}{\partial x} + \frac{\partial (z^2 - y^2)}{\partial y} + \frac{\partial (2yz + x^2)}{\partial z} \). This yields:\[ \abla \cdot \mathbf{V} = 3 - 0+ 0 + 2y\] Simplifying, we get \( \abla \cdot \mathbf{V} = 3 + 2y \).

Shift to Center the Sphere at Origin

The given sphere is \((x-2)^2 + (y+3)^2 + z^2 = 9\). To simplify integration, shift the origin: Let \(u = x-2\), \(v = y+3\), \(w = z\), hence the sphere becomes \(u^2 + v^2 + w^2 = 9\). In new coordinates, \( y = v - 3 \).

Change Vector Field Expression

Transform \(\mathbf{V}\) in terms of new coordinates after the shift: \(x = u + 2\), \(y = v - 3\), \(z = w\). Substituting yields \(\mathbf{V}= (3(u+2)-(v-3)w) \mathbf{i} + (w^2-(v-3)^2) \mathbf{j} + (2(w)(v-3)+(u+2)^2) \mathbf{k} \)

Adjust Divergence to New Coordinates

Next step is compute the volume integral of \(2v-3\) would be zero due symmetry about the origin i.e.. every contribution in one direction would be neutralized by a negative pair, i.e \[ \iiint_V 2v-3 \ dV= 0 \]

Key concepts

Surface Integral

A surface integral is a generalization of the concept of a line integral to surfaces. It is used to integrate a vector field over a surface. Imagine a surface like a sphere, a plane, or any other two-dimensional surface in three-dimensional space. A surface integral calculates how much of a vector field 'flows' through this surface.
In formal terms, the surface integral of a vector field \(\mathbf{V}\) over a surface \(S\) is denoted as \[ \iint_S \mathbf{V} \cdot \mathbf{n} \ d \sigma \] where \(\mathbf{n}\) is the unit normal vector to the surface and \(d \, \sigma\) is a small element of the surface area.
To appreciate this concept, think of a gentle breeze blowing across a beach ball. The surface integral would help us measure how much of the breeze penetrates into the ball, considering both the strength of the breeze and the orientation of the ball's surface.

Vector Field

A vector field is a construction in mathematics that associates a vector to every point in a subset of space. Think of it as assigning an arrow of specific length and direction to each point in an area. These arrows can represent quantities like velocity, force, or acceleration.
For example, in the given exercise, we are given the vector field \(\mathbf{V}=(3 x-y z) \mathbf{i}+\left(z^{2}-y^{2}\right) \mathbf{j}+\left(2 y z+x^{2}\right) \mathbf{k}\). Here, each component of \(\mathbf{V}\) represents the field's behavior in the x, y, and z directions.
Imagine a weather map showing wind velocities across a region. Each arrow on the map represents a vector, showing both the direction of the wind and its speed at that point.

Divergence

Divergence is a scalar measure of a vector field's 'spread' at a given point. It represents how much the vector field is diverging (spreading out) or converging (coming together) at that point.
The divergence of a vector field \(\mathbf{V}=(V_x, V_y, V_z)\) is given by the formula: \[ abla \cdot \mathbf{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z} \] This formula sums the rates at which the components of the vector field change along their respective axes.
In the step-by-step solution, we calculate the divergence of \(\mathbf{V}\) as: \[ abla \cdot \mathbf{V} = \frac{\partial (3x - yz)}{\partial x} + \frac{\partial (z^2 - y^2)}{\partial y} + \frac{\partial (2yz + x^2)}{\partial z} = 3 + 2y \] This value helps us understand the nature of the field’s 'spread' in the region.
Think of divergence like water flowing out of a hose. If water flows out uniformly from every point, it’s indicating divergence.