A field in physics is a quantity that's defined at every single point throughout a given space, allowing these quantities to vary from one point to another. This fundamental concept is crucial across various physics branches, from electromagnetism and gravity to fluid dynamics. Whether it's a scalar field assigning a single numerical value (like temperature) or a vector field assigning both a magnitude and direction (like wind velocity) to each point, fields are essentially a map from a "base space" to a space of values. They are vital for describing physical phenomena locally, meaning the value of a field at one point doesn't directly depend on values at other points—a property known as locality, which is a cornerstone of many physical theories. While the mathematics can seem complex, you encounter fields daily, like temperature maps or weather forecasts illustrating wind patterns.

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.

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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

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1. Understanding Vectors and Their Operations
2. Applications and Visualization of Cross Product Orthogonality
3. Vectors are Independent of Basis, Components Transform via Rotation Matrices
4. The Kronecker Delta and Permutation Symbol are Essential Tools for Vector Algebra and Geometric Interpretation
5. Fields as Functions Mapping Space to Physical Quantities
6. Integral Theorems: Connecting Derivatives to Boundaries
7. Vector Calculus in General and Orthogonal Coordinate Systems
8. Scalar and Vector Potentials: Decomposing Vector Fields and Their Properties

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1. Affine space and Coordinate System
2. 2D-the mass of a sphere with a position-dependant density
3. a gravitational acceleration vector at each point in space
4. the mass of a sphere with a given density field
5. integral force density over a volume
6. the electric field vectors around a point charge using Gauss’s law
7. The electric field vectors between positive and negative charge
8. A visual representation of the volume element and a numerical tool for calculating integrals of scalar and vector fields
9. 3D-Temperature Scalar Field
10. a pressure field within a 3D space </aside>