336 Vector Field

A mapping that assigns a vector to each point of a domain that is used to represent force-like quantities that vary with position.

definition [d] (Vector Field = Vector Function of Position) A function \(\mathbf{F}\) that assigns to each point \((x,y,z)\) in its domain a unique vector \(\mathbf{F}(x,y,z)\). In \(\mathbb{R}^{3}\):

• \(\mathbf{F}(x,y,z) = P(x,y,z)\, \mathbf{i} + Q(x,y,z)\, \mathbf{j} + R(x,y,z)\, \mathbf{k}\) .

where

• \(\mathbf{F}\) is the vector field.
• \(P, Q, R\) are scalar-valued component functions of position.
• \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the standard unit basis vectors.
• \((x,y,z)\) is a point in the domain.
• \(\mathbb{R}^{3}\) is three-dimensional Euclidean space.

Note:

• the vector is visualized as attached at the point \((x,y,z)\).

336.1 References

1. Stewart, J. Calculus. — vector field as a vector assigned to each point; component form \(P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}\).

• Basic Derivation of Jacobian
• Chain Rule
• Cross Product
• Curl
• Curve
• Definite Integral
• Derivation of Curl
• Derivation of Divergence
• Derivation of Grad
• Derivation of Jacobian
• Derivation of Taylor Expansion
• Divergence
• Dot Product
• Envelope of Tangent Lines
• General Functions
• Gradient
• Improper Integral
• Indefinite Integral
• Jacobian
• Legendre Transform
• Legendre Transform Derivation
• Limit
• Line Integral
• Notes
• Parameterization
• Partial Derivative
• Real Parameter
• Scalar Projection
• Smooth Curve
• Vector Calculus
• Vector Differential Operator
• Vector Field
• Vector Projection