Question

Prove the following identities. a. \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) (Hint: Apply the Divergence Theorem to each component of the identity.) b. \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\) (Hint: Apply Stokes' Theorem to each component of the identity.)

Short answer

Question: Prove the following vector calculus identities: a. \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) b. \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\) Answer: a. To prove the identity, we applied the Divergence Theorem to each component of the identity and used the property of curl and divergence. Finally, we related the volume and surface integral to prove the identity. b. To prove the identity, we applied Stokes' Theorem to each component of the identity and used the property of gradient and curl. Finally, we related the surface and line integral to prove the identity.

Step-by-step solution

Apply the Divergence Theorem for each component of the identity

The Divergence Theorem states that for a continuous vector field F and a closed surface S enclosing a volume D, we have: \(\iiint_{D} \nabla \cdot \mathbf{F} d V= \iint_{S} \mathbf{F} \cdot \mathbf{n} d S\). Now we apply the Divergence Theorem to each component of the identity: \((\mathbf{n} \times \mathbf{F})_i\).

Apply the property of curl and divergence

We know that \(\nabla \cdot (\nabla \times \mathbf{F}) = 0\). So the volume integral of the curl of a vector field F is zero: \(\iiint_{D}(\nabla \cdot (\nabla \times \mathbf{F}) ) dV = 0\).

Relate the volume and surface integral

Now, using the result from Step 1 and Step 2, we have: \(\iiint_{D} (\nabla \times \mathbf{F})_i dV =\iint_{S} (\mathbf{n} \times \mathbf{F})_i dS\). This is true for each component i. So, the identity is proved for the entire vector field: \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\). b. Proving the identity: \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\)

Apply Stokes' Theorem for each component of the identity

Stokes' Theorem states that for a continuous vector field on a surface S with a closed curve C as its boundary, we have: \(\iint_{S} (\nabla \times \mathbf{F}) \cdot \mathbf{n} d S = \oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Now we apply Stokes' Theorem to each component of the identity: \((\nabla \varphi)_i\).

Apply the property of gradient and curl

We know that \(\nabla \times (\nabla \varphi) = 0\). So, the surface integral of the cross product of the unit normal vector and the gradient of a scalar function φ is zero: \(\iint_{S}(\mathbf{n} \times \nabla \varphi)_i\cdot \mathbf{n} d S = 0\).

Relate the surface and line integral

Now, using the result from Step 1 and Step 2, we have: \(\oint_{C} \varphi d\mathbf{r}_i =\iint_{S} (\mathbf{n} \times \nabla \varphi)_i dS\). This is true for each component i. So, the identity is proved for the entire vector field: \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\). In conclusion, we have successfully proved both identities that were given in the exercise.

Key concepts

Divergence Theorem

The Divergence Theorem is a powerful tool in calculus, connecting the flow of a vector field across the boundary of a volume to the behavior of the field within the volume. It states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field over the region inside the surface. Mathematically, it can be expressed as:

\[\begin{equation}\iiint_{D} abla \cdot \mathbf{F} d V= \iint_{S} \mathbf{F} \cdot \mathbf{n} d S\end{equation}\]
Think of it as a way to total up all the 'source' amounts within a 3D shape by just looking at what crosses the boundary. This identity is especially useful in physics and engineering for calculating things like electric and fluid flow.

Stokes' Theorem

Stokes' Theorem is a statement about the integration of differential forms that provides a relationship between surface integrals and line integrals. Specifically, for a vector field \(\mathbf{F}\), Stokes' Theorem relates the circulation of \(\mathbf{F}\) around a closed curve to the curl of \(\mathbf{F}\) over the surface bounded by the curve:

\[\begin{equation}\iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} d S = \oint_{C} \mathbf{F} \cdot d \mathbf{r}\end{equation}\]
This theorem is extremely helpful when computing the work done by a field along a path or the amount of rotation (or curl) present along a surface. Essentially, it lets you transform a tough surface integral into a more manageable line integral.

Vector Fields

A vector field is a map that assigns a vector to each point in space. They represent quantities that have both magnitude and direction at every point, like the velocity of wind or the strength and direction of magnetic fields. A key visual image is to picture an arrow at every point in space, where the arrow's length and direction represent the vector's magnitude and direction. When we calculate line, surface, or volume integrals, we are aggregating these vectors in some way to understand the overall behavior of a physical system represented by the vector field.

Surface Integrals

Surface integrals extend the idea of a two-dimensional integral to functions over a curved plane or surface. If you have a scallop or patch we want to 'add up' a field across, we do so with a surface integral. Mathematically, a surface integral over surface \(S\) can be for a scalar field or vector field, represented by:

\[\begin{equation}\iint_S f dS\quad \text{or}\quad \iint_S \mathbf{F} \cdot \mathbf{n} dS\end{equation}\]
where \(\mathbf{n}\) is the unit normal vector to the surface. These integrals are crucial for calculating physical quantities like the electric flux or the force exerted by a pressure field across a surface.

Volume Integrals

Volume integrals are used when we want to accumulate a quantity over a three-dimensional region. Imagine filling up a space with tiny building blocks and adding up a quantity like mass or charge that each block contains. That's the essence of a volume integral. In formal terms, we write:

\[\begin{equation}\iiint_D f dV\end{equation}\]
for a function \(f\) over a region \(D\). Volume integrals are critical in many fields, for example, in calculating the total mass of an object with a variable density.

Line Integrals

Line integrals involve integration along a curve and they give us information about fields along paths. They can be used with scalar or vector fields, often to compute work done by a force field or to add up a field along a wire. For a vector field \(\mathbf{F}\) and a curve \(C\), the line integral is written as:

\[\begin{equation}\oint_C \mathbf{F} \cdot d\mathbf{r}\end{equation}\]
Where \(d\mathbf{r}\) represents a differential element of the curve. Line integrals are fundamental to understanding the circulation of fields and are closely related to physical concepts like work and energy.

Gradient

The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and its magnitude is the slope of the scalar field. The gradient is symbolized by \(abla f\) or \(\text{grad}\ f\), where \(f\) is a scalar function. It's a fundamental operator in vector calculus used to find how quantities change from one point to another in space. For example, in physics, the gradient of potential energy gives the force field experienced by a particle.

Curl of a Vector Field

The curl of a vector field is another vector field that represents the rotation or 'circulation' at each point. Its strength and direction signify the tendency of particles to rotate around that point. Mathematically, the curl of a vector field \(\mathbf{F}\) is given by \(abla \times \mathbf{F}\) and it's central to the study of fluid dynamics and electromagnetism. When the curl is zero at all points, a vector field is said to be irrotational, which has significant implications for the potential flows in various physical systems.