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物理代写|广义相对论代写General Relativity代考|PHY475

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物理代写|广义相对论代写General Relativity代考|Exterior derivative operator: generalised Stokes’ theorem

We have seen that (in an $n$-dimensional space) we may introduce 1-forms, with a corresponding line integral, and 2-forms, with a corresponding area integral-as well as 3-forms, with a volume integral, and so on. Consider, however, Stokes’ theorem
$$\int \mathbf{A} \cdot \mathrm{d} \mathbf{s}=\int \nabla \times \mathbf{A} \cdot \mathrm{d} \boldsymbol{\Sigma}$$

The left hand side is a line integral – the integral of a 1-form, while the right hand side is an area integral – the integral of a 2 -form. The very existence of a theorem like Stokes’ theorem implies that there must be a relation between 1-forms and 2-forms. In a similar way, Gauss’s theorem relates an integral over an area to one over a volume, implying a relation between 2 -forms and 3-forms. We now investigate this relation and find a beautiful generalisation of Stokes’ theorem, applicable to general forms, which yields both Stokes’ theorem and Gauss’s theorem as special cases.

The key to finding the relation is to define an exterior derivative operator $\mathbf{d}$ which converts a $p$-form into a $(p+1)$-form. Let $\omega$ be the $p$-form
$$\omega=a_{1, \ldots, p}(x) \mathbf{d} x^1 \wedge \cdots \wedge \mathbf{d} x^p .$$
then
$$\mathbf{d} \boldsymbol{\omega}=\frac{\partial a_{i_1 \cdots i_p}}{\partial x^k} \mathbf{d} x^k \wedge \mathbf{d} x^{i_1} \wedge \cdots \wedge \mathbf{d} x^{i_p} .$$

物理代写|广义相对论代写General Relativity代考|Generalised Stokes’ theorem

Let $\omega$ be a $p$-form, and let $c$ be a $(p+1)$-chain. Define $\partial$, the boundary operator on chains, so that $\partial c$ is the boundary of $c ; \partial c$ is a $p$-chain. Two simple examples are drawn in Fig. 3.11; in (a) $c$ is an area (2-chain) which is bounded by the closed line $\partial c$, a 1-chain. In (b) $c$ is a volume (3-chain) bounded by the surface $\partial c$, a (closed) 2-chain. Note that in both these cases the boundary is itself $c l o s e d ; \partial c$ has no boundary, or
$$\partial(\partial c)=\partial^2 c=0$$
This is a general result for $p$-chains:
$$\partial^2=0$$
and may be understood as being a result ‘dual’ to the Poincaré lemma $\mathbf{d}^2=0,(3.90)$ above. The boundary operator $\partial$ is dual to the exterior derivative operator $\mathbf{d}$.

Having defined the boundary operator we are now in a position to state the generalised Stokes’ theorem, which is
$$\int_{\partial c} \omega=\int_c d \omega .$$
Stokes’ theorem holds in any space, but to illustrate it let us work in $R^3$; and first consider the case $p=1$. Then $\omega$ is a 1-form, of the type (3.85), and $c$ is an area, with boundary $\partial c$, as in Fig. 3.11(a). The 2-form d $\omega$ is given by (3.86), where, as remarked already, the coefficients are the components of $\nabla \times \mathbf{a}$. Then (3.93) gives
$$\int_{\partial c} a \cdot \mathrm{d} l=\int_c(\nabla \times a) \cdot n \mathrm{~d} \Sigma$$
where $\mathrm{d} \boldsymbol{\Sigma}$ is an element of surface area, with unit normal $\mathbf{n}$. This is clearly Stokes’ theorem. As a second example take the case $p=2$, so $\omega$ is a 2-form, and therefore of the form (3.87); $\mathbf{d} \boldsymbol{\omega}$ is the 3 -form given by (3.88). The 3-chain $c$ is a volume $V$ with boundary $\partial c=\partial V$ (Fig. 3.11(b)) and (3.92) then gives
$$\int_{\partial V} b \cdot n \mathrm{~d} \Sigma=\int_V(\nabla \cdot b) \mathrm{d} V .$$
The reader will recognise this as Gauss’s theorem.
It will be appreciated that the generalised Stokes’ theorem is a neat and powerful theorem. The reader will doubtless recall that the ‘usual’ formulation of Stokes’ and Gauss’s theorems requires the stipulation of ‘directional’ notions – the normal $\mathbf{n}$ is an outward, not an inward, normal; and in Stokes’ theorem the path round the closed boundary is taken in an anticlock wise sense. These notions are however automatically encoded in the present formulation based on the exterior derivative operator, which, as we have seen, antisymmetrises and differentiates at the same time.

物理代写|广义相对论代写General Relativity代考|Exterior derivative operator: generalised Stokes’ theorem

$$\int \mathbf{A} \cdot \mathrm{d} \mathbf{s}=\int \nabla \times \mathbf{A} \cdot \mathrm{d} \boldsymbol{\Sigma}$$

$$\omega=a_{1, \ldots, p}(x) \mathbf{d} x^1 \wedge \cdots \wedge \mathbf{d} x^p .$$

$$\mathbf{d} \boldsymbol{\omega}=\frac{\partial a_{i_1 \cdots i_p}}{\partial x^k} \mathbf{d} x^k \wedge \mathbf{d} x^{i_1} \wedge \cdots \wedge \mathbf{d} x^{i_p} .$$

物理代写|广义相对论代写General Relativity代考|Generalised Stokes’ theorem

$$\partial(\partial c)=\partial^2 c=0$$

$$\partial^2=0$$

$$\int_{\partial c} \omega=\int_c d \omega .$$

$$\int_{\partial c} a \cdot \mathrm{d} l=\int_c(\nabla \times a) \cdot n \mathrm{~d} \Sigma$$

$$\int_{\partial V} b \cdot n \mathrm{~d} \Sigma=\int_V(\nabla \cdot b) \mathrm{d} V .$$

MATLAB代写

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