Elements of vector analysis

Definitions

Scalar field: application which associates a scalar (i.e. a number) with each point in space.

Vector field: application which associates a vector with each point in space.

Edges of volumes and surfaces: for a volume V, we note ∂V the surface delimiting this volume, oriented outwards (we also call it edge of V). Likewise, for an oriented (not closed) surface S, we note ∂S the contour “going around” this surface; its orientation depends on that of the surface (it is the edge of S). An example is given in Fig. 1.

Usual field characteristics

Level surface: for a scalar field f, set of points M such that there exists a constant k verifying f(M) ⸦ {k}

Field line: for a vector field , line L such that ∀M ∈ L, (M) is collinear with (M) or (M) is the tangent vector to L in M.

Fundamental quantities associated with a vector field

Circulation of a vector field: on an oriented contour C, C = ∫_C·d. More precisely, a contour is an application  : [0,1] → R³ and C = ∫₀¹((s)) (s)ds

Flow of a vector field: through an oriented surface S: Φ= ∫_C·d. A more precise definition involves a parameterization of the surface.

Fig. 1.1 – Oriented surface and its edge.

Fig. 1.2 – Cartesian, cylindrical and polar coordinates.

Locating a point in space

Cartesian coordinates: a point M is identified by its coordinates (x,y,z) such that  = x_x

+ y_y + z_z

Cylindrical coordinates: a point M is identified by its coordinates (r,θ,z) such that  = r_r + z_z

Spherical coordinates: a point M is identified by its coordinates (r,θ,φ) such that  = r_r

Source: Vincent Démery, Physique – 1ère année – 2ème année PSI. License CC BY-SA 2.0 FR. Translation and adaptation by Nicolae Sfetcu