Problem 4 - Calculus

Question

Compute the flux \(\iint \mathbf{F} \cdot \mathbf{n} d S\) by the Divergence Theorem. \(\mathbf{F}=x^{2} \mathbf{i}+8 \mathbf{y}^{2} \mathbf{j}+\boldsymbol{z}^{2} \mathbf{k}, \quad V ;\) unit cube.

Step-by-Step Solution

Answer

The flux is 10.

Step 1: Understand the Divergence Theorem

The Divergence Theorem relates the flux of a vector field through a closed surface to the divergence of the field inside the volume. Mathematically, it states that \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_V abla \cdot \mathbf{F} \, dV \), where \( S \) is the boundary of volume \( V \) and \( \mathbf{n} \) is the unit normal to the surface.

Step 2: Compute the Divergence

The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). For \( \mathbf{F} = x^2 \mathbf{i} + 8y^2 \mathbf{j} + z^2 \mathbf{k} \), the divergence is: \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(8y^2) + \frac{\partial}{\partial z}(z^2) = 2x + 16y + 2z \).

Step 3: Set Up the Integral Over the Volume

The unit cube has corners at \( (0,0,0) \) to \( (1,1,1) \). We need to integrate \( abla \cdot \mathbf{F} = 2x + 16y + 2z \) over this volume: \[ \iiint_V (2x + 16y + 2z) \, dx \, dy \, dz \].

Step 4: Integrate with Respect to x

Perform the integral with respect to \( x \) first: \[ \int_0^1 (2x + 16y + 2z) \, dx = \left[ x^2 + 16xy + 2xz \right]_0^1 = 1 + 16y + 2z \].

Step 5: Integrate with Respect to y

Now, integrate the resulting expression with respect to \( y \): \[ \int_0^1 (1 + 16y + 2z) \, dy = \left[ y + 8y^2 + 2yz \right]_0^1 = 1 + 8 + 2z = 9 + 2z \].

Step 6: Integrate with Respect to z

Perform the final integration with respect to \( z \): \[ \int_0^1 (9 + 2z) \, dz = \left[ 9z + z^2 \right]_0^1 = 9 + 1 = 10 \].

Step 7: Conclusion

The flux of \( \mathbf{F} \) through the surface of the unit cube using the Divergence Theorem is \( 10 \).

Key Concepts

Flux Calculation

Flux refers to how much of a vector field passes through a surface. Imagine you have water flowing through a net. The amount of water passing through can be thought of as flux. For vector fields, this is measured by the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \), where \( \mathbf{F} \) is the vector field and \( \mathbf{n} \) is a vector normal to the surface. This gives a precise way to calculate how much of the "field" is passing through a boundary.

• Calculating flux across closed surfaces can be simplified using the Divergence Theorem.
• This theorem converts a difficult surface integral into an easier volume integral.

By rewriting the surface flux in terms of the volume enclosed by the surface, we leverage the Divergence Theorem to transform a potentially complex calculation into a more straightforward one. This approach is particularly useful for complex shapes or when dealing with detailed vector fields.

Vector Fields

A vector field assigns a vector to each point in space. These vectors can represent various quantities like velocity of a fluid, gravitational fields, or even magnetic and electric fields. For example, the vector field \( \mathbf{F} = x^2 \mathbf{i} + 8y^2 \mathbf{j} + z^2 \mathbf{k} \) describes a field where each vector depends on its position in three-dimensional space.

• In this field, the 'i' component increases with \( x^2 \), meaning stronger influence as x increases.
• The 'j' component, linked to \( 8y^2 \), grows faster due to the coefficient 8, making changes in y more significant.
• The 'k' component varies with \( z^2 \), affecting the field vertically.

Understanding these components helps us comprehend how different parts of the field behave and how they contribute to the overall behavior of the field. This understanding is crucial when applying the Divergence Theorem to calculate flux across boundaries.

Volume Integration

Volume integration involves integrating a function over a three-dimensional region. In the context of the Divergence Theorem, it allows us to transform a surface integral into a volume integral. For our vector field, we compute the divergence \( abla \cdot \mathbf{F} \), which measures the rate of change or "spread" of the field within the volume.

• The divergence in our problem is given by \( abla \cdot \mathbf{F} = 2x + 16y + 2z \).
• We integrate this over the volume of the unit cube \( V \), with limits from \( (0,0,0) \) to \( (1,1,1) \).
• The integration is done across each dimension: \( x \), \( y \), and \( z \).

This step-by-step breakdown leads to the calculation of the total flux through the object, confirming that volume integration simplifies the process by condensing the information from the surface to its enclosed space, making the problem more manageable.

Surface Integrals

Surface integrals allow us to calculate the flux across a surface by integrating over that surface in three-dimensional space. More formally, a surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n} \, dS \) involves vectors and a surface area, considering how the vector field intersects each point on that surface.

• This type of integral can be intuitively understood as summing up the flow across a curved or flat boundary.
• Surface integrals take into account both the field's intensity and its direction relative to the surface.
• The direction of the normal vector \( \mathbf{n} \) is crucial as it defines what 'outward' flux means.

In our problem, the Divergence Theorem allows replacement of the surface integral with a volume integral, helping compute the otherwise challenging flux through the cube's entire surface with the divergence calculated within its volume.