Question

Use the divergence theorem to evaluate the surface integral $$\oiint_{S} u \cdot n d S$$ where \(u=\left(x \sin y, \cos ^{2} x, y^{2}-z \sin y\right)\) and \(S\) is the surface of the sphere \(x^{2}+y^{2}+(z-2)^{2}=1\)

Short answer

By applying the Divergence Theorem, the calculated divergence of \(u\) is zero, hence the surface integral \(\oiint_{S} u \cdot n d S\) equals zero.

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 within the volume enclosed by the surface. If \(S\) is a closed surface bounding a volume \(V\), then the Divergence Theorem states: \[\oiint_{S} \mathbf{u} \cdot d\mathbf{S} = \iiint_V (abla \cdot \mathbf{u}) \, dV\] where \(abla \cdot \mathbf{u}\) is the divergence of the field \(\mathbf{u}\).

Compute the Divergence of the Field

Find the divergence of the vector field \(u\). The divergence is given by \(abla \cdot u = \frac{\partial}{\partial x}(x \sin y) + \frac{\partial}{\partial y}(\(\cos^2 x\)) + \frac{\partial}{\partial z}(y^2 - z \sin y)\). Computation yields: \[abla \cdot u = \sin y + 0 - \sin y = 0\]

Apply the Divergence Theorem

Since the divergence of \(u\) is zero, the integral over the volume \(V\) will also be zero: \[\iiint_V (abla \cdot \mathbf{u}) \, dV = 0\] This implies that the surface integral over \(S\) is also zero.

Key concepts

Surface Integral

The surface integral is a fundamental concept in vector calculus, particularly when analyzing the flow of a vector field through a surface. Imagine a surface as a permeable membrane stretched across space, and a vector field represents a fluid flowing at various speeds and directions. A surface integral measures the total flow, or 'flux', of this fluid through the membrane.

Mathematically, the surface integral of a vector field \textbf{u} over a surface S is denoted as \[\oiint_{S} \textbf{u} \cdot d\textbf{S}\] which is the integral of the dot product of the vector field \textbf{u} and the differential area element of the surface, represented by the vector \(d\textbf{S}\). The direction of \(d\textbf{S}\) is generally taken to be the normal to the surface at each point, and its magnitude corresponds to the area of an infinitesimally small piece of the surface.

In practical terms, this concept is analogous to adding up how much fluid passes through each tiny piece of the permeable membrane, resulting in the total flux across the entire surface. This is an important concept in various fields of science and engineering where phenomena can be described using vector fields.

Vector Calculus

Vector calculus is an advanced mathematical tool that is used for analyzing and modeling physical phenomena in multiple dimensions. It extends the conventional calculus of single-variable functions to multivariable functions, covering operations including differentiation and integration over curves, surfaces, and volumes.

Vector fields are a key object of study in vector calculus, representing quantities that have both magnitude and direction at each point in space. Common operations with vector fields include taking their dot product, cross product, divergence, and curl.

Divergence and Curl

In particular, the divergence of a vector field can be roughly thought of as the rate at which 'fluid' expands or contracts at a point, while the curl measures the tendency to rotate.

Vector calculus is instrumental in physics, as it is used to describe fields such as gravitational, electric, and magnetic fields. It's also used in engineering disciplines such as fluid dynamics and electromagnetism. Understanding vector calculus is essential to grasp how quantities change and interact in space.

Flux of a Vector Field

The flux of a vector field through a surface can be visualized as the quantity of something (like liquid, gas, or even force lines) that passes through the surface. Technically, it is a surface integral that gives us this information.

For vector field \textbf{u} and surface S with an outward-pointing normal \textbf{n}, flux is defined using the dot product to capture the portion of the field that penetrates the surface perpendicularly: \[\text{Flux} = \oiint_{S} \textbf{u} \cdot \textbf{n} ds\] Where \(ds\) represents the differential area elements of S.

Flux plays a major role in understanding various physical principles, especially in electromagnetism and fluid dynamics. When the field represents electrical or magnetic fields, the flux calculates how much of the field lines pass through a given area, a concept integral to Maxwell's equations. Similarly, in fluid dynamics, flux measures the volume of fluid passing through an area over time, thus describing the flow rate through a section.