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# 物理代写|相对论代写Theory of relativity代考|FYS2004 TENSOR EQUATIONS

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## 物理代写|相对论代写Theory of relativity代考|TENSOR EQUATIONS

Introduction. We have seen that a (four-)vector is defined to be a quantity whose components transform in a specific way (i.e., like the components $d x^\mu$ of the displacement vector $d \mathbf{s}$ ) when we change our coordinate system:
$$A^{\prime \mu}=\frac{\partial x^{\prime \mu}}{\partial x^\nu} A^\nu$$
In this section, we will see how this definition fits into a larger scheme of quantities called tensors, and we will finally discover the fundamental reason why some component labels are written as superscripts and others as subscripts.

Covectors. A covector is defined to be a quantity whose components transform as follows when we change coordinate systems:
$$B_\nu^{\prime}=\frac{\partial x^\mu}{\partial x^{\prime \nu}} B_\mu$$
Compare equations $6.1$ and $6.2$ carefully. Note that the partial derivative for the covector transformation has the primed coordinate in the denominator of the partial derivative, not the numerator, and that the implied summation is over the coordinates in the numerator. Finally, note that in order to make the summation work, the index on the covector quantity must be a subscript, not a superscript. Putting a component label in the subscript position instead of the superscript position distinguishes covectors from ordinary vectors.

## 物理代写|相对论代写Theory of relativity代考|The Gradient as Covector

The Gradient as Covector. There are two important types of quantities that transform as covectors. The first is the gradient of a scalar function. Consider an arbitrary scalar function of position $\Phi(t, x, y, z) \equiv \Phi\left(x^\mu\right)$ whose value at any given event is frame-independent. The gradient of such a function is defined to be
$$\partial_\mu \Phi \equiv \frac{\partial \Phi}{\partial x^\mu}$$
This object has one component for each of the possible values of the free index $\mu$ (so in 4D spacetime, it has four components). According to the chain rule of partial differential calculus, the values of these components transform as follows:
$$\partial_\nu^{\prime} \Phi \equiv \frac{\partial \Phi}{\partial x^{\prime \nu}}=\frac{\partial x^\mu}{\partial x^{\nu \nu}} \frac{\partial \Phi}{\partial x^\mu}=\frac{\partial x^\mu}{\partial x^{\nu \nu}} \partial_\mu \Phi$$
If you compare this with equation $6.2$, you will see that the components of the gradient do transform as the components of a covector. Box $6.1$ discusses simple 2D examples of gradient covectors.

## 物理代写|相对论代写THEORY OF RELATIVITY代考|TENSOR EQUATIONS

$$A^{\prime \mu}=\frac{\partial x^{\prime \mu}}{\partial x^\nu} A^\nu$$

$$B_\nu^{\prime}=\frac{\partial x^\mu}{\partial x^{\prime \nu}} B_\mu$$

## 物理代写|相对论代写THEORY OF RELATIVITY代考|THE GRADIENT AS COVECTOR

$$\partial_\mu \Phi \equiv \frac{\partial \Phi}{\partial x^\mu}$$

$$\partial_\nu^{\prime} \Phi \equiv \frac{\partial \Phi}{\partial x^{\prime \nu}}=\frac{\partial x^\mu}{\partial x^{\nu \nu}} \frac{\partial \Phi}{\partial x^\mu}=\frac{\partial x^\mu}{\partial x^{\nu \nu}} \partial_\mu \Phi$$

## Matlab代写

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