Question

\(\iiint \nabla \cdot V d \tau\) over the volume \(x^{2}+y^{2} \leq 4,0 \leq z \leq 5, V=\left(\sqrt{x^{2}+y^{2}}\right)(\hat{i x}+\hat{j})\)

Short answer

20\pi

Step-by-step solution

- 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 enclosed by the surface. It states \[ \iiint_V (abla \cdot V) \, d\tau = \iint_{S} V \cdot dS \].

- Calculate the Divergence of V

The vector field is given as \( V = \sqrt{x^2 + y^2}( \hat{i} + \hat{j} ) \). The divergence is defined by \( abla \cdot V = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z} \). For \( V = \sqrt{x^2 + y^2}( \hat{i} + \hat{j} ) \), \( V_x = \sqrt{x^2 + y^2} \) and \( V_y = \sqrt{x^2 + y^2} \), and \( V_z = 0 \). Compute the partial derivatives: \[ \frac{\partial V_x}{\partial x} = \frac{\partial}{\partial x} \left( \sqrt{x^2 + y^2} \right) = \frac{x}{\sqrt{x^2 + y^2}}, \] and similarly, \[ \frac{\partial V_y}{\partial y} = \frac{y}{\sqrt{x^2 + y^2}}. \] Therefore, \( abla \cdot V = \frac{x}{\sqrt{x^2 + y^2}} + \frac{y}{\sqrt{x^2 + y^2}} = 1. \)

- Integrate the Divergence over the Volume

Since \( abla \cdot V = 1 \), the integral over the given volume becomes \[ \iiint_{V} 1 \, d\tau. \] This is simply the volume of the given region. The given volume bounds are a cylinder with radius 2 and height 5. The volume of a cylinder is given by \( \pi r^2 h \) where \( r = 2 \) and \( h = 5 \). Hence, the volume is \[ \pi (2^2)(5) = 20\pi. \]

Key concepts

Vector Calculus

Vector calculus is a field of mathematics concerned with vector fields and operations on them. It's widely used in physics and engineering to describe and study physical phenomena like fluid flow, electromagnetic fields, and much more.

The fundamental operations in vector calculus include differentiation and integration of vector fields. This includes concepts like the gradient, curl, and divergence, each providing different kinds of insights into the behavior and properties of vector fields.

By understanding these operations, we are able to analyze and solve problems involving physical quantities that have both magnitude and direction.

Divergence of a Vector Field

The divergence of a vector field is a scalar measure of a vector field’s tendency to originate from or converge into a point. In simple terms, it tells us how much a vector field is 'spreading out' or 'converging' at any given point.

Mathematically, the divergence of a vector field \( V \) is given by: \( abla abla \text {cdot} V = \frac{\text {partial} V_x}{\text {partial} x} + \frac{\text {partial} V_y}{\text {partial} y} + \frac{\text {partial} V_z}{\frac{\text {partial} z}} \).

In our exercise, we had \( V = \text {sqrt} x^2 + y^2( \text {hat} i + \text {hat} j ) \), and calculating its divergence gives us: '1', which means every point in the field is uniformly spreading out.

Volume Integral

A volume integral is the integral of a function over a 3-dimensional region (volume). In the context of the Divergence Theorem, it is crucial because it allows us to relate the divergence of a vector field within a volume to the flux through the surface that encloses this volume.

For our particular problem, given that the divergence of the vector field within the volume is 1, the volume integral of the divergence over the given cylindrical volume is essentially computing the volume of the cylinder. This is why our task simplified to calculating \( iiint_V 1 d \tau \).

Finally, recognizing that the volume is simply a cylinder with radius 2 and height 5, the integral computation results in \( 20\text {pi} \).

Cylindrical Volume

Understanding the volume of a cylinder is essential when dealing with problems in vector calculus, especially those involving cylindrical symmetry.

A cylinder's volume is calculated using the formula: \( \text {pi } r^2 h, \) where \( r \) is the radius of the base and \( h \) is the height of the cylinder.

In our given problem, the region described by \( x^2 + y^2 leq 4, 0 dec z leq 5 \) outlines a cylinder with radius
2 and height 5.

Applying the volume formula, we get \( \text {pi } (2^2) (5 ) = 20\text {pi} \). This volume is crucial because integrating the divergence over this region essentially required us to compute this volume.