Question

Miscellaneous integral identities 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 integral identities using the given theorems: (a) \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) using the Divergence Theorem. (b) \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\) using Stokes' Theorem. Answer: (a) Apply the Divergence Theorem to each component of the given identity and add up the obtained equations to get the desired identity. (b) Apply Stokes' Theorem to the gradient of the scalar function \(\varphi\), and notice that the resulting equation is equivalent to the given identity.

Step-by-step solution

Identity (a)

To prove the identity \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\), we will use the Divergence Theorem, which states that \(\iiint_{D} \nabla \cdot \mathbf{G} dV = \iint_{S} (\mathbf{n} \cdot \mathbf{G}) dS\). We are given the hint to apply it to each component of the identity. Let \(\mathbf{G} = (\nabla \times \mathbf{F})_x \hat{i}\) and apply the Divergence Theorem: \begin{align*} \iiint_{D} \nabla \cdot \mathbf{G} dV &= \iint_{S} (\mathbf{n} \cdot \mathbf{G}) dS \\ \iiint_{D} \nabla \cdot ((\nabla \times \mathbf{F})_x \hat{i}) dV &= \iint_{S} (\mathbf{n} \cdot ((\nabla \times \mathbf{F})_x \hat{i})) dS \end{align*} Repeat this process for \(\mathbf{G} = (\nabla \times \mathbf{F})_y \hat{j}\) and \(\mathbf{G} = (\nabla \times \mathbf{F})_z \hat{k}\). Adding up the three obtained equations, we get the desired identity.

Identity (b)

To prove the identity \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\), we will use Stokes' Theorem, which states that \(\iint_{S} (\nabla \times \mathbf{G}) \cdot \mathbf{n} dS = \oint_{C} \mathbf{G} \cdot d\mathbf{r}\). We are given the hint to apply it to each component of the identity. Let \(\mathbf{G} = \nabla \varphi\), and apply Stokes' Theorem: \begin{align*} \iint_{S} (\nabla \times \mathbf{G}) \cdot \mathbf{n} dS &= \oint_{C} \mathbf{G} \cdot d\mathbf{r} \\ \iint_{S} (\nabla \times (\nabla \varphi)) \cdot \mathbf{n} dS &= \oint_{C} (\nabla \varphi) \cdot d\mathbf{r} \end{align*} Now notice that \(\nabla \times (\nabla \varphi) = \mathbf{n} \times (\nabla \varphi)\). Thus, the equation we derived is equivalent to the identity we are trying to prove: \begin{align*} \iint_{S} (\mathbf{n} \times \nabla \varphi) d S &= \oint_{C} \varphi d \mathbf{r} \end{align*}

Key concepts

Divergence Theorem

The Divergence Theorem is a fundamental result in vector calculus that relates the flow (divergence) of a vector field through a closed surface to the behavior of the vector field inside the volume enclosed by the surface. In simpler terms, it allows us to calculate the 'outflow' of a vector field from a volume by integrating over the surface that bounds the volume. The formal statement is:
\[ \iiint_{D} abla \cdot \mathbf{G} \, dV = \iint_{S} (\mathbf{n} \cdot \mathbf{G}) \, dS \]
where \( \mathbf{G} \) is a continuously differentiable vector field defined on a volume \( D \), \( S \) is the boundary of \( D \), and \( \mathbf{n} \) is the outward-facing unit normal vector on \( S \). This theorem is especially useful when we need to compute a volume integral and converting it to a surface integral can simplify the calculation. In the given exercise, the Divergence Theorem is utilized to prove an identity involving the curl of a vector field \( \mathbf{F} \), by treating each component of the curl as a divergence.

Stokes' Theorem

Stokes' Theorem is another central theorem in vector calculus that gives a relationship between a surface integral and a line integral. It essentially generalizes the Fundamental Theorem of Calculus to higher dimensions. The theorem states that the circulation of a vector field \( \mathbf{G} \) around a closed loop \( C \) is equal to the total curl of \( \mathbf{G} \) over the surface \( S \) that is bounded by \( C \). The formal statement of the theorem is:
\[ \iint_{S} (abla \times \mathbf{G}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{G} \cdot d\mathbf{r} \]
Here, \( abla \times \mathbf{G} \) represents the curl of \( \mathbf{G} \), and \( \mathbf{n} \) is the unit normal vector to the surface \( S \), pointed in the direction according to the right-hand rule. By using Stokes' Theorem, one can derive identities or solve integrals involving rotations of fields around curves, as demonstrated in the exercise where the theorem helps prove an identity involving the cross product of the unit normal \( \mathbf{n} \) and the gradient of a scalar function \( \varphi \).

Vector Calculus

Vector calculus is a branch of mathematics that deals with the differentiation and integration of vector fields, often in two or three-dimensional Euclidean space. It includes operations such as gradient, curl, and divergence, which measure the rate of change of scalar and vector fields. These operators are essential tools for analyzing physical quantities such as velocity, force, and magnetic and electric fields.
For example, the gradient (denoted by \( abla \varphi \)) represents the rate of change of a scalar field and points in the direction of the greatest increase of the function. On the other hand, the curl (denoted by \( abla \times \mathbf{F} \)) measures the twist or rotation of a vector field around a point. Lastly, the divergence (denoted by \( abla \cdot \mathbf{F} \)) measures the magnitude of a source or sink at a given point in a vector field. These concepts not only appear in mathematical theory but are also integral in fields such as physics, engineering, and computer graphics.

Multivariable Calculus

Multivariable calculus is the extension of calculus to functions of more than one variable. While single-variable calculus is concerned with functions that depend on one variable, multivariable calculus deals with functions depending on two or more variables, like \( f(x, y) \) or \( g(x, y, z) \). It encompasses the study of partial derivatives, multiple integrals, and the aforementioned vector calculus operations.
Partial derivatives represent the rate of change of functions with respect to one variable while keeping the other variables constant. Multiple integrals, such as double and triple integrals, generalize the idea of integration to calculate volumes and other quantities in higher dimensions. Multivariable calculus is fundamental in analyzing and solving problems in areas that require the modeling of three-dimensional space and beyond, including many problems in physics, economics, and engineering.