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## Substance

*A summary of Penrose's summary of physics and math.*

- Argues that thus far in dealing with Riemann surfaces via holomorphic functions, we have only been working in one dimension: the complex dimension
- these surfaces then constitute complex curves
- complex conjugation is the "appropriate" way to approach splitting the complex parameter z into its real and imaginary parts
- this is a non-holomorphic operation
The complex conjugate of \(z = x+iy\) is \(\bar{z}=x-iy\)
\(\bullet\) in the complex \(z\)-plane this operation constitutes a reflection of the plane in the real line By including the complex conjugate, we can get \(x\) and \(y\) by: \[x = \frac{z+\bar{z}}{2}\] \[y = \frac{z-\bar{z}}{2i}\] we can then express functions \(f(x, y)\) as \(F(z, \bar{z})\). These are 2-real-dimensional and non-holomorphic. Suppose you have a 2D surface \(S\) and a smooth function \(\Phi\) defined on \(S\). \(\Phi\) can then be a smooth map from \(S\) to \(\mathbb{R}\) or \(\mathbb{C}\). \(\bullet\) such a \(\Phi\) may be called a scalar field on \(S\) \(\bullet\) for \(\Phi = f(x, y)\) to be smooth, it must have both the derivatives wrt. \(x\) and \(y\) to be continuous functions of the coordinate pair \((x, y)\) \(\bullet\) for \(C^{1}\)-smoothness \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\) must be continuous \(\bullet\) for \(C^{2}\)-smoothness \(\frac{\partial^{2}f}{\partial x^{2}}\), \(\frac{\partial^{2}f}{\partial y^{2}}\) and \(\frac{\partial^{2}f}{\partial x\partial y}\) must be continuous \(\bullet\) for \(C^{3}\)-smoothness: \[\frac{\partial^{3}f}{\partial x^{3}}\] \[\frac{\partial^{3}f}{\partial y^{3}}\] \[\frac{\partial^{3}f}{\partial x^{2}\partial y} = \frac{\partial^{3}f}{\partial x \partial y \partial x} = \frac{\partial^{3}f}{\partial y\partial x^{2}}\] and presumably \(\frac{\partial^{3}f}{\partial x\partial y^{2}}\) must all be continuous \(\bullet\) should we have a second set of coordinates on which \(\Phi\) acts, we have also \(\Phi = F(X, Y)\) "On an overlap region between the two patches, we shall therefore have \(f(x, y) = F(X, Y)\)." (pg. 185) \(X\) and \(Y\) in this case could be fucntions of \(x\) and \(y\). These are called transition fucntions: \[X = X(x, y) ... Y = Y(x, y)\] \[x = x(X, Y) ... y = y(X, Y)\] \(\bullet\) when the transition functions are smooth, the smoothness of \(\Phi\) is independent of the choice of coordinates on the overlapping region ## Vector Fields and 1-Forms
"There is a notion of 'derivative' of a function that is independent of the coordinate choice." (pg. 185)
\(\bullet\) for function \(\Phi\) defined on a 2-manifold surface \(S\), it is denoted \(d\Phi\) (which is a 1-form) \[d\Phi = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy\] in "disembodied operator form" this is \(d = dx\frac{\partial}{\partial x} + dy\frac{\partial}{\partial y}\) which arises from the chain rule. The operator \(\frac{\partial}{\partial x}\) which acts on the fucntion \(\Phi\) defined on the 2-manifold \(S\) can be given meaning through the chain rule which can relate differentiation of the variables \(X\) and \(Y\) to \(x\) as \[\frac{\partial}{\partial x} = \frac{\partial X}{\partial x}\frac{\partial}{\partial X} + \frac{\partial Y}{\partial x}\frac{\partial}{\partial Y}\] \(\bullet\) a vector field on \(S\) in the \((X, Y)\)-coordinate patch is defined as \[\mathcal{E} = A\frac{\partial}{\partial X} + B\frac{\partial}{\partial Y}\] where \(A\) and \(B\) are shown above.
In the \((x, y)\)-coordinate patch (or system) we have similarly \[\mathcal{E} = a\frac{\partial}{\partial x}+b\frac{\partial}{\partial y}\] can think of this as the gradient of \(\Phi\) \(\star\) note that even if \(X = x\), \(\frac{\partial}{\partial x} \neq \frac{\partial}{\partial X}\) since there is also a dependence on the other coordinate \(\bullet\) \(d\Phi\) = gradient/ exterior derivative of \(\Phi\) ---can be used to obtain contour lines on \(S\) ---\(\frac{\partial}{\partial x}\) points along lines of constant \(y\) ---\(\frac{\partial}{\partial y}\) points along lines of constant \(x\) \(a\) and \(b\) can be used to label \(\mathcal{E}\), referred to as its components in the \((x, y)\) system (or coordinate labels) \(\bullet\) \(d\Phi = udx + vdy\) where \(u\) and \(v\) are components of \(d\Phi\) in \((x, y)\) system \(\bullet\) \((u, v)\) of \(d\Phi\) (the 1-form) and \((a, b)\) of \(\mathcal{E}\) (the vector field) are related by the scalar or inner product (\(\mathcal{E}(\Phi)\) or \(d\Phi\bullet\mathcal{E}\): \[\mathcal{E}(\Phi) = d\Phi\bullet\mathcal{E} = au + bv = a\frac{\partial\Phi}{\partial x} + b\frac{\partial\Phi}{\partial y}\] \(\bullet\) a 1-form is also called a covector \(\bullet\) 1-forms are dual to vector fields \(\bullet\) contour lines occur where \(d\Phi\bullet\mathcal{E} = 0\) ## The Cauchy-Riemann Equations
How to characterize the complex-valued functions \(\Phi\) which are holomorphic?
\(\bullet\) on (\(x\), \(y\)), \(\Phi\) must be holomorphic in \(z = x + iy\) \(\bullet\) on (\(X\), \(Y\)), \(\Phi\) must be holomorphic in \(z = X + iY\) \(\bullet\) \(Z\) must be a holomorphic function of \(z\) where the coordinate systems overlap in \(S\) Suppose we can express \(\Phi\) as a function of \(z\) and \(\bar{z}\) and that \(\Phi\) is independent of \(\bar{z}\), that is \(\frac{\partial\Phi}{\partial\bar{z}} = 0\), then \(\Phi\) is holomorphic. By the chain rule, this is equivalent to \(\frac{\partial\Phi}{\partial x} + i\frac{\partial\Phi}{\partial y} = 0\). Writing \(\Phi = \alpha + i\beta\) we can get \(\frac{\partial\alpha}{\partial z} = \frac{\partial\beta}{\partial y}\) and \(\frac{\partial\alpha}{\partial y}=-\frac{\partial\beta}{\partial x}\), the Cauchy-Riemann equations. Then we must also have \[\frac{\partial X}{\partial x} = \frac{\partial Y}{\partial y}\] and \[\frac{\partial X}{\partial y} = -\frac{\partial Y}{\partial x}\] when this holds for any (\(x\), \(y\)) and (\(X\), \(Y\)) we have a Riemann surface \(S\). When \(\alpha\) and \(\beta\) satisfy \(\frac{\partial\alpha}{\partial x} = \frac{\partial\beta}{\partial y}\) and \(\frac{\partial\alpha}{\partial y} = -\frac{\partial\beta}{\partial x}\) then Laplace's equations are satisfied \[\nabla^{2}\alpha = 0 ... \nabla^{2}\beta = 0\] the 2-dimensional Laplacian is defined \[\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}\] Laplace's equation in 3D plays fundamental role in \(\bullet\) Newtonian gravity: equation satisfied by potential function determining the gravitational field \(\bullet\) electrostatics: equation satisfied by the potential function determining the static electric field (both in free space) \(\star\) solutions to the Cauchy-Riemann equations can be derived from solutions to teh 2D Lapalce equation if \(\alpha\) satisfies \(\nabla^{2}\alpha=0\), then we get \(\beta\) from \(\beta = \int\frac{\partial\alpha}{\partial x}dy\)
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