Notation for a "scalar absolute field"?

In summary, the notation used to describe a vector field V in a 3D space is F = ||∫^space_s|V|ds||, where vectors in opposite directions do not cancel and are instead added to the magnitude. This can be represented in Python code using numpy arrays. For example, for a 2D space with two forces acting on a point, the absolute value of the vector force at that point can be found by adding the absolute values of the individual forces and normalizing it to a scalar value.
  • #1
timjdoom
6
3
TL;DR Summary
I have a mathematical expression which I'm unsure of the formal representation or name of. The best way I can think to describe it is as a "scalar absolute field".
The notation I think best describes it is

## F = \lVert\int^{space}_s|\vec{V}|ds\rVert ##

So you have a vector field V in a 3d space. For each point you integrate over all of space (similar to a gravitational or electromagnetic field) *but* vectors in opposite directions do not cancel, they add to the magnitude (i.e. you integrate the absolute of the vector). Then finally you want the scalar value, so you normalise the vector.

For context the python (numpy) code to express it is given say a 4-dimensional array `vec` (a 3D vector in 3D space) might be

Python:
xyz = np.array([np.sum(dimension) for dimension in np.abs(vec)])
field = np.sum(xyz**2, axis=0)**0.5

Example for gravitation (or charge) in a 2D space
- You have a mass ##M_a## at position a, exerting a force ##\vec{F_a} = -3\hat{x} +3\hat{y}## on point p.
- Next you have a force ##M_b## at position b, exerting a force ##\vec{F_b} = +2\hat{x} +2\hat{y}## on point p.
- The vector force at point p is therefore ##\vec{F_p} = \vec{F_a}+\vec{F_b} = -1\hat{x} +5\hat{y}##
- Meaning the absolute of the vector is ##|\vec{F_p}| = 1\hat{x} + 5\hat{y}##, and normalising would give you the scalar of ##\lVert\vec{F_p}\rVert = \sqrt{1^2+5^2} = \sqrt{26}##
- What I need is ##|\vec{F_a}|+|\vec{F_b}| = (|-3|+|2|)\hat{x}+ (|3|+|2|)\hat{y} = 5\hat{x}+5\hat{y}##, which gives you the scalar ##\lVert|\vec{F_p}|\rVert = \sqrt{5^2+5^2} = \sqrt{50}##
 
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  • #2
Just integrate the scalar rho/r2 for gravity, and whatever other magnitude you have for other forces? No need to invent more notation.
 

Related to Notation for a "scalar absolute field"?

1. What is a "scalar absolute field"?

A scalar absolute field is a mathematical concept used in physics and engineering to describe a quantity that has both magnitude and direction. It is a scalar because it only has magnitude and not direction, and it is absolute because it is independent of any coordinate system.

2. How is a scalar absolute field represented in notation?

A scalar absolute field is typically represented by a bold lowercase letter, such as f, with a subscript to indicate the specific field. For example, fx would represent the scalar absolute field in the x-direction.

3. What is the difference between a scalar absolute field and a vector field?

A scalar absolute field is a quantity that has magnitude but no direction, while a vector field has both magnitude and direction. In other words, a scalar absolute field is a scalar quantity, while a vector field is a vector quantity.

4. How is a scalar absolute field calculated?

The value of a scalar absolute field at a specific point in space is calculated by taking the absolute value of the scalar field at that point. This value is independent of any coordinate system and is the same regardless of the direction of measurement.

5. What are some real-world applications of scalar absolute fields?

Scalar absolute fields are used in many fields of science and engineering, such as fluid dynamics, electromagnetism, and thermodynamics. They are particularly useful in situations where the direction of a quantity is not important, such as temperature, pressure, or concentration. They are also used in computer graphics and image processing to represent grayscale images.

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