Review: Calculus in Hyperspace

38.1 GEOMETRIES

38.1.1 \(\mathbb{R}^{4}\) Essentials

The four dimensional Euclidean space \(\mathbb{R}^{4}=M(4,1)\) is the space of column vectors with four real components \(X=[x, y, z, w]^{T}\). If we think of such a vector as a point, we also write \(X=(x, y, z, w)\). The dot product \(=\) inner product allows as usual to define length \(|X|=\sqrt{X \cdot X}\), the distance \(|X-Y|\) and the angles \(\cos (\alpha)=(X \cdot Y) /(|X||Y|)\) between vectors. The Cartesian coordinate system has now four axes which are perpendicular to each other. Historically, as \(\mathbb{R}^{4}\) is also the space of quaternions, it is custom to label the coordinate directions as \[1=[1,0,0,0], \quad i=[0,1,0,0], \quad j=[0,0,1,0], \quad k=[0,0,0,1].\] A vector \([3,4,5,1]\) for example is then written also as \(3+4 i+5 j+k\). We will however keep the vector-form. We will come back in the last section of this document about why quaternions are natural.

38.1.2 Dimensionality of Kernels

The kernel of the \(1 \times 4\) matrix \(A=[a, b, c, d]\) defines the linear hyperplane \[a x+b y+c z+d w=0.\] It is a \(3\)-dimensional linear space. An example is the coordinate hyperplane \(x=0\), which consists of all points \(\{(0, y, z, w) \mid y, z, w \in \mathbb{R}\}\). More generally, the solution space \[a x+b y+d z+d w=e\] is an affine hyperplane. The kernel of a \(2 \times 4\) matrix is in general, as an intersection of two hyperplanes, a \(2\)-dimensional plane, which we just call a plane. The kernel of a \(3 \times 4\) matrix \(A\) is in general a line. Geometrically, it is the intersection of three hyperplanes.

38.1.3 Cruising Hyperspace: From Spheres to Hyper-Tori

A symmetric \(4 \times 4\) matrix \(B\), a row vector \(A \in M(1,4)\) and a constant \(e\) define the hyper quadric \(X \cdot B X+A X=e\). For a diagonal matrix \(B=\operatorname{Diag}(a, b, c, d)\), this gives the quadric \[a x^{2}+b y^{2}+c z^{2}+d w^{2}=e.\] Examples are the \(\boldsymbol{3}\)-sphere \(x^{2}+y^{2}+z^{2}+w^{2}=1\), the hyper paraboloid \(x^{2}+y^{2}+z^{2}=w\), the \(3\)-cylinder \(x^{2}+y^{2}+z^{2}=1\) which is the product of a \(2\)-sphere and a line. Or the cylinder-plane \(x^{2}+y^{2}=1\) which can be seen as the product of the \(1\)-sphere with a \(2\)-plane. There are three types of hyperboloids like \[\begin{aligned} x^{2}+y^{2}+z^{2}-w^{2}&=1,\\ x^{2}+y^{2}-z^{2}-w^{2}&=1,\\ x^{2}-y^{2}-z^{2}-w^{2}&=1. \end{aligned}\] One could call them \(\boldsymbol{1}\)-hyper-hyperboloids, \(\boldsymbol{2}\)-hyper-hyperboloids and \(\boldsymbol{3}\)-hyper-hyperboloids, using the Morse index as a label. There is still \(\boldsymbol{1}\)-hyperbolic-paraboloid \(x^{2}+y^{2}-z^{2}=w\) but there are more degenerate surfaces like \(x^{2}-y^{2}=w\). The two-dimensional torus \(\mathbb{T}^{2}\) can be realized here as a quadratic surface. It is the intersection of \(x^{2}+y^{2}=1\), \(z^{2}+w^{2}=1\). This is the flat torus. We can not realize the two-dimensional torus in a flat way in our three dimensional space \(\mathbb{R}^{3}\). In hyper-space, it can. There is also a three dimensional torus \(\mathbb{T}^{3}\). To get a parametrization, start with the \(2\)-torus parametrization \[r(\phi, \theta)=\big[(3+\cos (\phi)) \cos (\theta),(3+\cos (\phi)) \sin (\theta), \sin (\phi)\big]\] then expand the circle to get a hyper-torus \[\begin{aligned} r(\phi, \theta, \psi)=\Big[&(3+\cos (\phi)) \cos (\theta),(3+\cos (\phi)) \sin (\theta),\\ &(3+\sin (\phi)) \cos (\psi),(3+\sin (\phi)) \sin (\psi)\Big]^{T}. \end{aligned}\] You see that for every fixed \(\psi\) we have a \(2\)-torus. We can compute \[4|d r|=18+6 \cos (\phi)+6 \sin (\phi)+\sin (2 \phi)\] which is always positive and so verifies that the map from \(\mathbb{T}^{3}\) to \(\mathbb{R}^{4}\) is locally injective. We can also easily check that if \(\psi\) or \(\theta\) is fixed we get a translated scaled version of the \(2\)-torus. If \(\phi\) is fixed, we get the flat \(2\)-torus mentioned above.

38.1.4 From Curves to Hyper-Surfaces: Seeing in Hyperspace

In single variable calculus, one looks at graphs \(\{(x, y) \mid y=f(x)\}\) of functions of one variable. In multi-variable, one adds graphs \(\{(x, y, z) \mid z=f(x, y)\}\) of functions of two variables. The graph of a function \(w=f(x, y, z)\) is now a \(3\)-dimensional space. Paraboloids like \(w=x^{2}+y^{2}+z^{2}\) or \(w=x^{2}+y^{2}-z^{2}\) are graphs. An other example is the three dimensional bell hyper-surface \[w=f(x, y, z)=\pi^{-3 / 2} e^{-x^{2}-y^{2}+z^{2}},\] where the constant has been chosen so that the hyper-volume \(0 \leq w \leq f(x, y, z)\) is equal to \(1\). For obvious reasons, we usually do not draw the graph of a function of three variables as we would have to draw in \(4\) dimensions. Now, in hyperspace, we can do that.

38.1.5 Higher-Dimensional Parametrization

Spaces can be parametrized in the same way as we parametrized curves or surfaces in three dimensions. A curve is defined by four real functions \(x(t)\), \(y(t)\), \(z(t)\), \(w(t)\) of one variables and written as \[r(t)=[x(t), y(t), z(t), w(t)]^{T}.\] A surface is parametrized by \[r(u, v)=[(x(u, v), y(u, v), z(u, v), w(u, v)].\] A hypersurface is now defined by \[r(u, v, t)=[x(u, v, t), y(u, v, t), z(u, v, t), w(u, v, t)].\]

38.1.6 Transformations in \(\mathbb{R}^{4}\)

A coordinate change is defined by a map from \(\mathbb{R}^{4}\) to \(\mathbb{R}^{4}\) given by four differentiable functions: \[r(u, v, s, t)=\big[x(u, v, s, t), y(u, v, s, t), z(u, v, s, t), w(u, v, s, t)\big].\] We have seen already the parametrization \[r(\phi, \theta_{1}, \theta_{0})=[\cos (\phi) \cos (\theta_{1}), \cos (\phi) \sin (\theta_{1})\sin (\phi) \cos (\theta_{2}), \sin (\phi) \sin (\theta_{2})]\] of the unit \(\boldsymbol{3}\)-sphere \(=\) hyper-sphere \[x^{2}+y^{2}+z^{2}+w^{2}=1.\] Because \(z=x^{2}+y^{2}+z^{2}\) is a cylinder, there is also a natural cylindrical coordinate system in four dimensions. It is given by \[r(\rho, \phi, \theta, w)=[\rho \sin (\phi) \cos (\theta), \rho \sin (\phi) \sin (\theta), \rho \cos (\phi), w].\] If we write down the Jacobian matrix and compute the determinant we get \(\rho^{2} \sin (\phi)\) as in spherical coordinates.

38.2 FIELDS

38.2.1 Differential Forms in \(\mathbb{R}^{4}\)

A scalar function \(f(x, y, z, w)\) is also called a \(0\)-form. A vector field is denoted by \(F=[P, Q, R, S]^{T}\) and a \(\boldsymbol{1}\)-form \(F=[P, Q, R, S]\) is written as \[F=P \,d x+Q \,d y+R \,d z+S \,d w.\] A \(\boldsymbol{2}\)-form \(F\) has \(6\) components: \[F=A \,d x \,d y+B \,d x \,d z+C \,d x \,d w+P \,d y \,d z+Q \,d y \,d z+R \,d z \,d w.\] A \(\boldsymbol{3}\)-form again has four components \[P \,d y \,d z \,d w+Q \,d x \,d z \,d w+R \,d x \,d y \,d w+S \,d x \,d y \,d z\] and a \(\boldsymbol{4}\)-form is again completely determined by a scalar function \(f\) because \[F=f \,d x \,d y \,d z \,dw.\]

38.2.2 Exterior Derivatives of Forms

The exterior derivatives are computed by using the anti-commutation rule like \[d x \,dy=-d y \,dx \quad \text{and} \quad d f=f_{x} \,dx+f_{y} \,dy+f_{z} \,dz+f_{w} \,dw\] and extending this to terms like \[\begin{aligned} P \,dy \,dz&=d P \,dy \,dz\\ &=(P_{x} \,dx+P_{y} \,dy+P_{z} \,dz+P_{w} \,dw) \,dy \,dz\\ &=P_{x} \,dx \,dy \,dz+P_{w} \,dw \,dy \,dz. \end{aligned}\] For a \(1\)-form \[F=P \,dx+Q \,dy+R \,dz+S \,dw\] we have \[\begin{aligned} d F&=P_{x} \,dx \,dx+P_{y} \,dy \,dx+P_{z} \,dz \,dx+P_{w} \,dw \,dx\\ &\quad +Q_{x} \,dx \,dy+Q_{y} \,dy \,dy+Q_{z} \,dz \,dy+Q_{w} \,dw \,dy\\ &\quad +R_{x} \,dx \,dz+R_{y} \,dy \,dz+R_{z} \,dz \,dz+R_{w} \,dw \,dz\\ &\quad +S_{x} \,dx \,dw+S_{y} \,dy \,dw+S_{z} \,dz \,dw+S_{w} \,dw \,dw \end{aligned}\] which simplifies to expression with \(6\) terms. We have because every term like \(P_{y z} \,dz \,dy \,dx\) is paire dwith a term like \(P_{z y} \,dy \,dz \,dx\) which cancel. For a \(2\)-form \[\begin{aligned} F&=A \,dx \,dy+B \,dx \,dz+C \,dx \,dw\\ &\quad +P \,dy \,dz+Q \,dy \,dw\\ &\quad +R \,dz \,dw, \end{aligned}\] we have \[\begin{aligned} d F&=(A_{z} \,dz+A_{w} \,dw) \,dx \,dy+(B_{y} \,dy+B_{w} \,dw) \,dx\,dz\\ &\quad +(C_{y} \,dy+C_{z} \,dz) \,dw \,dx+(P_{x} \,dx+P_{w} \,dw) \,dy \,dz\\ &\quad +(Q_{x} \,dx+Q_{z} \,dz) \,dy \,dw+(R_{x} \,dx+R_{y} \,dy) \,dz \,dw \end{aligned}\] which simplifies to \[\begin{aligned} dF &=(Q_{z}+P_{w}+R_{y}) \,dy \,dz \,dw+(B_{w}+C_{z}+R_{x}) \,dx \,dz \,dw\\ &\quad +(A_{w}+Q_{x}+C_{y}) \,dx \,dy \,dw+(A_{z}+B_{y}+P_{x}) \,dx \,dy \,dz. \end{aligned}\] For a \(3\)-form \[F=P \,dy \,dz \,dw+Q \,dz \,dw \,dx+R \,dw \,dx \,dy+S \,dx \,dy \,dz\] we have \[d F=(P_{x}-Q_{y}+R_{z}-S_{w}) \,dx \,dy \,dz \,dw.\]

38.2.3 Differential Operators on Fields

The gradient of a function \(f(x, y, z, w)\) is defined as \[\nabla f(x, y, z, w)=d f^{T}=[f_{x}, f_{y}, f_{z}, f_{w}]^{T}.\] The curl of a vector field \(F(x, y, z, w)=[F_{1}, F_{2}, F_{3}, F_{4}]^{T}\) is the hyperfield \[d F=[F_{12}, F_{13}, F_{14}, F_{23}, F_{24}, F_{34}]^{T},\] where we have just chosen a lexigographic order and where \(F_{i j}=\partial_{x_{j}} F_{i}-\partial_{x_{i}} F_{j}\). The hypercurl of a hyper vector field \[F(x, y, z, w)= [F_{12}, F_{13}, F_{14}, F_{21}, F_{23}, F_{34}]\] is a \(3\)-form but can again be associated with a vector field \[d F=[F_{234}, F_{134}, F_{124}, F_{123}]^{T}.\] The divergence of a vector field \(F=[P, Q, R, S]\) is a \(4\)-form \[(P_{x}+Q_{y}+R_{z}+S_{w}) \,dx \,dy \,dz \,dw\] but can again be associated with a scalar field.

38.2.4 Relationships Between Differential Operators

Here are some properties which we have seen already. The gradient \(\nabla f=d f^{T}\) is perpendicular to the level surface \(f(x, y, z, w)=c\). The curl of the gradient is zero. The hypercurl of the curl is zero. The divergence of the hypercurl is zero. The divergence of the gradient is the Laplacian (using the identifications, the divergence map can be identified with the adjoint \(-d^{*}\)). The chain rule is \[d / d t f(r(t))=\nabla f(r(t)) \cdot r^{\prime}(t).\]

38.2.5 Integration in \(\mathbb{R}^{4}\)

The line integral of a vector field \(F\) along a curve \(C\) is \[\int_{C} F(r(t)) \cdot r^{\prime}(t) \,d t.\] The flux integral of a vector field \(F\) along a \(2\)-dimensional surface is a flux integral. The hyper flux integral of a hyper-field \(F\) along a surface. The hyper volume integral of a function \(f\) on a solid \(G\) is \[\iiiint_{G} f(x, y, z, w) \,dx \,dy \,dz \,dw.\]

38.3 THEOREMS

38.3.1 The Fundamental Theorem for Line Integrals in \(\mathbb{R}^{4}\)

The fundamental theorem of line integrals is:

38.3.2 Stokes Theorem

The Stokes theorem tells that for a surface \(S\) and \(1\)-form \(F\):

38.3.3 Higher-Dimensional Stokes Theorem

The Hyper Stokes theorem assures that for a hypersurface \(S\) and a \(2\)-form \(F\), the flux of the hypercurl of \(F\) through \(G\) (a \(3\)D-integral) is the flux of \(F\) through the boundary surface \(S\) (a \(2\)D-integral):

38.3.4 Divergence Theorem

The divergence theorem assures that for a \(3\)-form (identified as a vector field \(F\)) and a solid \(G\) with boundary hyper-surface \(S\), we have:

38.4 QUATERNIONS

38.4.1 Lie Groups: From Dough to Particles

Hyperspace \(\mathbb{R}^{4}\) is special: it is the only Euclidean space for which the unit sphere is a non-Abelian Lie group. A Lie group \(G\) is a manifold¹ \(r(\mathbb{R}^{m}) \subset \mathbb{R}^{n}\) on which one has a group operation \(x * y\) which has the property that for every \(y\), the maps \(x \rightarrow x * y\) and \(x \rightarrow y * x\) are smooth maps on \(G\). To have a group \((G, *)\) we must have the property that \((x * y) * z=x *(y * z)\) and that there is a \(\boldsymbol{1}\)-element \(1 * x=x * 1=x\) such that every element \(x\) has an inverse \(x^{-1}\) satisfying \(x * x^{-1}=1\). The circle \[\{x^{2}+y^{2}=1\}=\{z \in \mathbb{C} \mid |z|=1\}\] is an example of a group. This multiplication is Abelian if \(x * y=y * x\) for all \(x, y \in G\). The complex plane \(\mathbb{C}=\mathbb{R}^{2}\) is characterized as the only Euclidean space \(\mathbb{R}^{n}\) in which the unit sphere \(\mathbb{T}^{1}=\{|x|=1\}\) is an Abelian Lie group. Why Lie groups? They are the dough, elementary particles are baked from! Electromagnetism is built from \(\mathbb{T}^{1}\) for example.

38.4.2 From Vectors to a Division Algebra

One can write a vector in \(\mathbb{R}^{4}\) also as \[v=a+i b+j c+k d\] where \(i\), \(j\), \(k\) are symbols. Hamilton noticed that when defining \[i^{2}=j^{2}=k^{2}=i j k=-1,\] the \(4\)-dimensional space becomes an algebra. An algebra is a linear space which also features a multiplication. Now one has already \(M(2,2)\), the space of \(2 \times 2\) matrices, which is a \(4\)-dimensional algebra, but the algebra which Hamilton found is a division algebra: every non-zero element can be inverted. This is not the case for \(M(2,2)\). The matrix in which all elements are \(1\) for example is non zero but it is also not invertible.

38.4.3 Quaternion Basics: Conjugation and Norm

The algebra which Hamilton defined through the relations \[i^{2}=j^{2}=k^{2}=i j k=-1\] is called the quaternion algebra \(\mathbb{H}\). If \[\bar{v}=a-i b-j c-k d,\] then \(|v|^{2}=v \cdot v=v \bar{v}\), where the right hand side is a quaternion multiplication. One can readily check that \(|v w|=|v||w|\). The reason is that quaternions \(v\) can be realized as complex \(2 \times 2\) matrices: if \[A(v)=\left[\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right],\] then \(|v|=\operatorname{det}(A(v))\) and \(A(v) A(w)=A(v w)\). Your favorite AI helps to check this last identity quickly.

38.4.4 Division Algebras

An algebra with the property \(|v * w|=|v||w|\) is a normed division algebra. By theorems of Hurwitz and Frobenius, there are only four: the reals \(\mathbb{R}\), the complex \(\mathbb{C}\), the quaternions \(\mathbb{H}\) and the octonions \(\mathbb{O}\). For an associative division algebra, the unit sphere is a Lie group. Because the unit sphere of \(\mathbb{R}\) has only two points, the \(1\)-circle \(\{|z|=1\} \subset \mathbb{C}\) and the unit \(3\)-sphere \(\{|z|=1\} \subset \mathbb{H}\) are the only spheres that are Lie groups. There is a unique non-commutative one, the \(3\)-sphere and a unique commutative connected one, the \(1\)-sphere.