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A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$ (note that is a $(k+1)$-form)

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

Applies Einstein's summation convention and you will have the desired result.

A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

Applies Einstein's summation convention and you will have the desired result.

A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$ (note that is a $(k+1)$-form)

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

Applies Einstein's summation convention and you will have the desired result.

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A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

Applies Einstein's summation convention and you will have the desired result.

A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

Applies Einstein's summation convention and you will have the desired result.

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A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

A exterior derivative (the measure of how much an $k$-form varies along an $n$-dimensional manifold) for an $k$-form $\phi=g dx^A = gdx^i \wedge ...\wedge dx^p$ (where $g$ is a 0-form i.e., a smooth function) with $A$ as an index set defined by $A=\{i_{n}\}_{n=1}^{k}$ with $i_{k}=p$, is written in the following form:

$d\phi=\frac{\partial g}{\partial x_i}dx^i \wedge dx^A$

e.g. if $\phi$ is an 0-form defined by $\phi=g(x)$ the exterior derivative is

$d\phi=\frac{\partial g(x)}{\partial x_{i}}dx^i$

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