Question

\(\iint(2 x \mathbf{i}-2 y \mathbf{j}+5 \mathbf{k}) \cdot \mathbf{n} d \sigma\) over the surface of a sphere of radius 2 and center at the origin.

Short answer

The value of the surface integral is 0.

Step-by-step solution

Understand the Problem

The given exercise involves calculating the surface integral of a vector field \((2x \, \mathbf{i} - 2y \, \mathbf{j} + 5 \, \mathbf{k}) \cdot \mathbf{n} \, d\sigma\) over the surface of a sphere. The sphere has a radius of 2 and is centered at the origin.

Apply the Divergence Theorem

To simplify the problem, apply the Divergence Theorem. The Divergence Theorem states that \(\iint_S \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_V (abla \cdot \mathbf{F}) \, dV\), where \(\mathbf{F} = 2x \, \mathbf{i} - 2y \, \mathbf{j} + 5 \ k\) and \(abla \cdot \mathbf{F}\) is the divergence of \(\mathbf{F}\).

Compute the Divergence of the Vector Field

Calculate the divergence \(abla \cdot \mathbf{F}\). The divergence is given by: \((abla \cdot \mathbf{F}) = \frac{\partial (2x)}{\partial x} + \frac{\partial (-2y)}{\partial y} + \frac{\partial (5)}{\partial z} = 2 - 2 + 0 = 0\).

Evaluate the Volume Integral

Since the divergence of the vector field is zero, the volume integral \(\iiint_V (abla \cdot \mathbf{F}) \, dV\) over the volume of the sphere is zero. Therefore, \((\iiint_V 0 \, dV = 0)\).

Conclude the Result

By the Divergence Theorem, the surface integral \(\iint_S (2x \, \mathbf{i} - 2y \, \mathbf{j} + 5 \, \mathbf{k}) \cdot \mathbf{n} \, d\sigma\) is equal to the volume integral, which we found to be zero.

Key concepts

surface integrals

Surface integrals are used to calculate the flux of a vector field across a given surface. In simpler terms, they measure how much of the vector field is passing through the surface. For this, we need the vector field and the normal vector to the surface.
The given problem involves the surface integral of a vector field \((2x \mathbf{i} - 2y \mathbf{j} + 5 \mathbf{k}) \cdot \mathbf{n} d\sigma\) over a sphere. Here, \mathbf{n} is the unit normal vector, and \sigma denotes the surface area.

When applying the surface integral, we consider:

• The vector field: This is given by \(2x \mathbf{i} - 2y \mathbf{j} + 5 \mathbf{k}\).
• The surface: In this case, it's a sphere of radius 2 centered at the origin.
• The normal vector \mathbf{n}\: For a sphere centered at the origin, this is simply the normalized position vector.

Surface integrals can be quite complex, which is why the Divergence Theorem is often used to simplify them.

vector fields

A vector field assigns a vector to every point in space. These vectors can represent various physical quantities like force, velocity, or magnetic fields. In the given problem, our vector field is expressed as:

\((2x \mathbf{i} - 2y \mathbf{j} + 5 \mathbf{k})\)

Here, each vector (located at a point \((x, y, z)\) in space) is represented by its three components:

• It's made up of the \mathbf{i}, \mathbf{j}, and \mathbf{k\text{-} directions with magnitudes 2x, -2y, and 5 respectively.
• The \mathbf{i} direction points along the x-axis.
• The \mathbf{j} direction points along the y-axis.
• The \mathbf{k} direction points up along the z-axis.

Understanding vector fields is crucial when delving into surface and volume integrals, as we often need to calculate how the field interacts with or passes through a certain surface or volume.

volume integrals

Volume integrals allow us to calculate quantities over three-dimensional regions in space. They are particularly useful when applying the Divergence Theorem. According to the theorem:

\[ \iint_S \mathbf{F} \cdot \mathbf{n} \, d\sigma = \iiint_V \left( abla \cdot \mathbf{F} \right) \, dV \]

This means that the surface integral of a vector field across a closed surface can be transformed into a volume integral over the region contained within the surface.

• In our problem, the surface is a sphere with a radius of 2.
• The corresponding volume is the space inside this sphere.
• To apply the Divergence Theorem, we need to compute the divergence of the field \( \mathbf{F} = 2x \mathbf{i} - 2y \mathbf{j} + 5 \mathbf{k}\).

In this problem, since the divergence computed is zero:

\[ abla \cdot \mathbf{F} = \frac{\partial (2x)}{\partial x} + \frac{\partial (-2y)}{\partial y} + \frac{\partial (5)}{\partial z} = 2 - 2 + 0 = 0 \]

The volume integral over the region inside the sphere is:

\[ \iiint_V ( abla \cdot \mathbf{F} ) \, dV = \iiint_V 0 \, dV = 0 \]

This leads us to the conclusion that the original surface integral is also zero.