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## Main Question or Discussion Point

Reading Roger Penrose's

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Covariant derivative:

[tex](1)\enspace \nabla (\alpha + \beta)=\nabla \alpha + \nabla \beta[/tex]

[tex](2)\enspace \nabla (\alpha \cdot \beta)= (\nabla \alpha) \cdot \beta + \alpha \cdot (\nabla \beta)[/tex]

[tex](3)\enspace \nabla \phi = \mathrm{d}\phi=\frac{\partial \phi}{\partial x^i} \mathrm{d}x^i[/tex]

(The dot in (2) stands for a contraction of some number of pairs of indices, including possibly a contraction of no indices, i.e. a tensor product.)

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Exterior derivative:

[tex](1)\enspace \mathrm{d}(\alpha + \beta)=\mathrm{d} \alpha + \mathrm{d} \beta[/tex]

[tex](2)\enspace \mathrm{d}(\alpha \wedge \beta)=\mathrm{d} \alpha \wedge \beta + (-1)^p \, \alpha \wedge \mathrm{d} \beta[/tex]

[tex](3)\enspace \mathrm{d}(\mathrm{d} \alpha)=0[/tex]

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I guess one difference would be that in general [itex]\nabla (\nabla \phi) \neq 0 = \mathrm{d}(\mathrm{d}\phi)[/itex].

*The Road to Reality*, I wondered what is the relationship/difference between these? Can one be expressed simply in terms of the other? The exterior derivative seems to be only defined for form fields. He says the covariant derivative of a scalar field (0-form field) is the same as its exterior derivative. Is this the case for higher dimensional forms, or can the covariant derivative of a form field be different from its exterior derivative? The rules Penrose gives for them look different on some points, but are there differences that aren't accounted for by the antisymmetric property of forms? In the following, alpha and beta are tensors, phi a scalar.*

Covariant derivative:

[tex](1)\enspace \nabla (\alpha + \beta)=\nabla \alpha + \nabla \beta[/tex]

[tex](2)\enspace \nabla (\alpha \cdot \beta)= (\nabla \alpha) \cdot \beta + \alpha \cdot (\nabla \beta)[/tex]

[tex](3)\enspace \nabla \phi = \mathrm{d}\phi=\frac{\partial \phi}{\partial x^i} \mathrm{d}x^i[/tex]

(The dot in (2) stands for a contraction of some number of pairs of indices, including possibly a contraction of no indices, i.e. a tensor product.)

*

Exterior derivative:

[tex](1)\enspace \mathrm{d}(\alpha + \beta)=\mathrm{d} \alpha + \mathrm{d} \beta[/tex]

[tex](2)\enspace \mathrm{d}(\alpha \wedge \beta)=\mathrm{d} \alpha \wedge \beta + (-1)^p \, \alpha \wedge \mathrm{d} \beta[/tex]

[tex](3)\enspace \mathrm{d}(\mathrm{d} \alpha)=0[/tex]

*

I guess one difference would be that in general [itex]\nabla (\nabla \phi) \neq 0 = \mathrm{d}(\mathrm{d}\phi)[/itex].

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