Fields and multivariable functions

A field assigns a value to every point

A scalar field such as temperature gives one number at each position and time; a vector field such as velocity gives both magnitude and direction. The field viewpoint replaces tracking every particle with a function over space.

Move from samples to a function

Write a scalar as φ(x,y,z,t). A small displacement changes it by dφ = φ_x dx + φ_y dy + φ_z dz + φ_t dt. The partial derivatives hold other coordinates fixed; they are local slopes that can be measured from nearby values.

A temperature field T(x,y) = 20 + 3x − 2y °C has T_x = 3 °C/m and T_y = −2 °C/m. Moving 0.1 m east and 0.2 m north changes temperature by approximately 0.3 − 0.4 = −0.1°C.

A field is only as good as its domain

A continuum field smooths over microscopic variation and may fail near shocks, interfaces, or discrete particles. State whether coordinates are physical position, material labels, or a transformed computational grid.

Work it: For φ = x²y + 4t, calculate partial derivatives at (x,y,t) = (2,3,1). Estimate Δφ for dx = 0.01 and dy = −0.02 with dt = 0.