Fields in physics are best understood as functions mapping points in space (and time) to physical quantities. Operations on fields, like multiplication, are performed locally; the value of a resulting field at any point only depends on the input fields' values at that same point. Fields can also be integrated over lines, surfaces, or volumes within their base space to yield total quantities.

<aside> 🥅

Synthesizing an excerpt is crucial for grasping a discipline's multifaceted nature.

🎬Animated result and interactive web

$\gg$🧠Cloud AI for numerical analysis and code verification

</aside>

More precisely:

• A field assigns a value (which may be a scalar, vector, or tensor) to every point in space and time, i.e., it is a function

  $$ \phi: \text { space (or spacetime) } \rightarrow \text { physical quantity. } $$

For example, the electric field $\vec{E}(\vec{r}, t)$ assigns a vector (electric field strength and direction) to each point $\vec{r}$ in space and time.

• The physical quantities are the values the field takes at specific points. The field as a whole is the function, while the measurable physical quantity is the field's value at a given location (and time). For instance, measuring the temperature at one location yields a scalar physical quantity; the entire temperature field is the function that gives temperature everywhere.
• Fields can be of different types:
  • Scalar fields: assign a single number to every point, e.g., temperature or atmospheric pressure.
  • Vector fields: assign a vector to every point, e.g., velocity of fluid flow, electric or magnetic fields.
  • Tensor fields: assign more complex quantities like tensors, useful in describing stress or curvature in space.
• In classical field theory, fields are continuous functions defined over space and time with infinitely many degrees of freedom, contrasting with classical particle mechanics which deals with finitely many degrees of freedom indexed by particle labels.
• In quantum field theory, fields are elevated to operator-valued functions of space and time, describing fundamental quantum degrees of freedom.

The field itself is the mapping or function that assigns a physical quantity (scalar/vector/tensor) to every point in spacetime, and the physical quantity is the value of the field at a particular point. This mathematical abstraction allows physicists to describe spatially and temporally varying quantities across physical systems consistently and is central both in classical and quantum physics.

Cloud computing facilitates diverse field visualizations and analyses, offering animated results for concepts like affine spaces and charge-generated electric fields, interactive web tools for understanding volume elements and 3D scalar/vector fields, and plotting capabilities for showcasing density gradients, surface normals, and various vector field properties.

Fields as Functions Mapping Space to Physical Quantities

Synthesizing an excerpt is crucial for grasping a discipline's multifaceted nature.

<aside> 🥠

Understanding Vectors and Their Operations-1

Applications and Visualization of Cross Product Orthogonality-2

Vectors are Independent of Basis, Components Transform via Rotation Matrices-3

The Kronecker Delta and Permutation Symbol are Essential Tools for Vector Algebra and Geometric Interpretation-4

Fields as Functions Mapping Space to Physical Quantities-5

</aside>

🎬Animated result and interactive web

Affine space and Coordinate System

Affine space and Coordinate System

2D the mass of a sphere with a position dependent density #physics #python

2D-the mass of a sphere with a position-dependant density

a gravitational acceleration vector at each point in space

a gravitational acceleration vector at each point in space

the mass of a sphere with a given density field

the mass of a sphere with a given density field

Integral force density over a volume

integral force density over a volume