Are the existence of matter and energy life's ONLY mysteries?

Cypress said:
:lolup::lolup:
The gravitational, electric, and magnetic fields are all vector fields. :palm:

You don't get to overturn 200 years of physics. In physics, a field by definition assigns a value to every point in space. The only way to describe the gravitational field, the magnetic field, or the Higgs field is to assign a numerical value to every point in space.
Click to expand...
Why EVERY point?

Why not just the points about which we give a shit
and then just round up figuring for the rest of them?

The absolute pinnacle of my math skills
was learning how to calculate the On-Base Plus Slugging stat (OPS).
Thus, there could conceivably be a flaw in my reasoning, here.
 
NiftyNiblick said:
Why not just the points about which we give a shit
Click to expand...
^^ That is the basic idea.
In principle, it's just like a map of temperatures at the earth's surface, which itself is like a grid or mathematical scalar field. Every point in space on the Earth's surface in theory has a unique temperature, but the National weather service might only have a few dozen temperature monitoring stations in a county or region. That is good enough for what we need, but we understand that in theory a few dozen thermometers does not capture all the temperature variation at arbitrarily smaller scales.
 
Cypress said:
That is the basic idea.
In principle, it's just like a map of temperatures at the earth's surface, which itself is like a grid or mathematical scalar field. Every point in space on the Earth's surface in theory has a unique temperature, but the National weather service might only have a few dozen temperature monitoring stations in a county or region. That is good enough for what we need, but we understand that in theory a few dozen thermometers does not capture all the temperature variation at arbitrarily smaller scales.
Click to expand...
That makes sense, of course.

I may not be a genius,
but I'm not so dim as to not know
how much I don't know!
 
NiftyNiblick said:
Why EVERY point?

Why not just the points about which we give a shit
and then just round up figuring for the rest of them?

The absolute pinnacle of my math skills
was learning how to calculate the On-Base Plus Slugging stat (OPS).
Thus, there could conceivably be a flaw in my reasoning, here.
Click to expand...

AI Overview
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Imaginary numbers, specifically complex numbers, can be visualized and used in the context of vector fields. Complex numbers can be represented as vectors in the complex plane, where the real part corresponds to the x-axis and the imaginary part to the y-axis. This representation allows for operations like addition and scalar multiplication to be visualized as vector operations.


Visualizing Complex Numbers as Vectors:
  • A complex number, such as a + bi, can be plotted on the complex plane as a point or vector, where 'a' is the real part and 'b' is the imaginary part.

  • The length of the vector from the origin to this point represents the modulus (or magnitude) of the complex number, and the angle it makes with the real axis represents its argument.

  • Adding two complex numbers corresponds to adding the vectors they represent, following the parallelogram rule for vector addition.

  • Multiplying a complex number by a scalar (real number) stretches or shrinks the vector proportionally to the scalar value.

  • Multiplying a complex number by i (the imaginary unit) rotates the vector by 90 degrees counterclockwise.
Using Complex Numbers in Vector Fields:

    • In some fields, complex numbers provide a more convenient and elegant way to represent and analyze vector fields, particularly in scenarios involving rotations and oscillations.
  • For example, in electrical engineering, complex numbers are used to represent alternating currents (AC) and voltages, where the imaginary part represents the phase shift.
  • In fluid dynamics, complex potentials can be used to analyze two-dimensional, steady, irrotational flows.
  • The use of complex numbers in vector fields can lead to a more compact and insightful representation of physical phenomena.
Key Differences and Considerations:

    • While complex numbers can be represented as vectors, they also possess a multiplication structure that vectors do not.

  • The addition of complex numbers corresponds to vector addition, but the multiplication of complex numbers is more complex and involves both scaling and rotation.

  • The concept of a field is different from a vector field. A field is an algebraic structure with operations like addition and multiplication, while a vector field assigns a vector to each point in a space.

  • In some cases, complex analysis, which deals with functions of complex variables, can offer a more powerful framework than traditional vector analysis.
 
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