How is gradient a one form?

June 10, 2020

How is gradient a one form?

One form is a tensor or a rule that takes one vector as an argument and returns a real number. Simply it is a mapping from vector to real number. Gradient is a commonly known one form. To understand how a gradient is a one form, consider the following diagram.

Suppose 𝜙( $\vec{x}$ ) is a scalar field defined at every event $\vec{x}$ . Its value changes from event to event. The scalar field is a function of coordinates of events.
$\phi (\vec{x}) = \phi[t, x , y , z ]$

The curve shown in the figure is the world line of a particle. At every point of that curve the scalar field 𝜙( $\vec{x}$ ) gives some real number value. t is the time for a particular event measured by the observer 𝓞. Now consider the proper time 𝜏 which is measured by the particle itself in its frame. If we label all the events by the proper time 𝜏, you can treat the coordinates of the events on its world line as scalar functions of 𝜏.
$[ t = t(\tau ), x = x(\tau), y = y(\tau), z = z(\tau)]$
For every value of 𝜏, there are real number values of those coordinates. As the coordinates themselves are functions of the proper time 𝜏, 𝜙 is also a function of proper time 𝜏.
$\phi (\tau) = \phi[ t = t(\tau ), x = x(\tau), y = y(\tau), z = z(\tau)]$

Now if we want to determine the rate of change of the scalar field 𝜙 along the curve, we have to find the value of d𝜙/d𝜏.

$\frac{d\phi}{d\tau} = \frac{\delta \phi}{\delta t}\frac{dt}{d\tau} + \frac{\delta \phi}{\delta x}\frac{dx}{d\tau} + \frac{\delta \phi}{\delta y}\frac{dy}{d\tau} + \frac{\delta \phi}{\delta z}\frac{dz}{d\tau}$

$\Rightarrow \frac{d\phi}{d\tau} = \frac{\delta \phi}{\delta t}U^{t} + \frac{\delta \phi}{\delta x}U^{x} + \frac{\delta \phi}{\delta y}U^{y} + \frac{\delta \phi}{\delta z}U^{z}$

It is clear from the last equation that from the vector $\vec{U}$ we get the number d𝜙/d𝜏 which represents the rate of change of 𝜙 on a curve on which $\vec{U}$ is tangent. As you can see the number d𝜙/d𝜏 is a linear function of $\vec{U}$ , so we have defined a one-form. The one form has components (𝛿𝜙/𝛿t,𝛿𝜙/𝛿x,𝛿𝜙/𝛿y,𝛿𝜙/𝛿z). This one form is called the gradient of 𝜙 denoted by d̃𝜙:

This gradient satisfies all the axioms of one form. Note that the rate of change of the scalar field 𝜙 is a scalar product of d𝜙/d𝑥⃗ and d𝑥⃗/d𝜏. So,
$\Rightarrow \frac{d\phi}{d\tau} = \frac{d \phi}{d \vec{x}}\frac{d\vec{x}}{d\tau}$
(from partial derivative we can say that).The scalar product is a rule that takes two vectors and gives a real number. If only one vector is passed to the scalar product rule then it still lacks one vector. This can be treated as a one form rule that takes a vector and makes the scalar product between the two vectors. For an example, suppose a rule g̃( $\vec{A}$ , $\vec{B}$ ) = $\vec{A}$ . $\vec{B}$ . If we pass only one vector g̃( $\vec{A}$ , ) lacks one. So, g̃( $\vec{A}$ , )can be treated as an another rule Ã( ) which is a one form and takes only one vector and is defined by Ã( $\vec{B}$ ) = $\vec{A}$ . $\vec{B}$ . Similarly, d̃𝜙 is a one form that takes a vector as an argument and gives a real value which is the scalar product of d𝜙/d𝑥⃗(called the gradient vector) and the vector. To find the rate of change of 𝜙 along the curve we have to pass the tangent four velocity vector $\vec{U}$ as an argument to the gradient d̃𝜙. It is very much important for you to understand that d̃𝜙 itself is not the change of the scalar field 𝜙 along the curve on which $\vec{U}$ is tangent. It is just a rule that takes the vector $\vec{U}$ and gives a real number which is the rate of change of 𝜙 along the curve. So, you can see that the gradient is actually a one-form.

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How is gradient a one form?

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