Is there such a thing as a scalar absolute field?

Question

I have a mathematical expression which I feel must be a better / more formal representation.

The best way I can think to describe it is as a "scalar absolute field" (but there maybe a better name for it).

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

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

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 $\lVert|\vec{F_p}|\rVert = \sqrt{5^2+5^2} = \sqrt{50}$

Answer 1

Define ,$\vec{F}=\sum_{i}^{n} {F_{i}}\vec{e_{i}}$ We define the relation, for for vectors as , $|\vec{F}|=\bigoplus_{i=1}^{n}|{F_{i}}|\vec{e_{i}}$ where absolute value defined by op. Now, tried to think of giving a structure it is indeed possible.Now go on as you wish, Next for continuous case you can think of like breaking the component in Riemann Sum or Some integrable sum and apply the direct sum unless you cannot define it in continuous case , now think to give it a new notation (cool)!!! Best Regards!!!