Vector Analysis

Scalar

A physical quantity that has magnitude but no direction (e.g., temperature, mass, time).

Vector

A physical quantity that has both magnitude and direction (e.g., force, velocity, displacement).

Scalar Field

A function that assigns a scalar value to each point in space (e.g., temperature distribution in a room).

Vector Field

A function that assigns a vector to each point in space (e.g., velocity of fluid particles, gravitational field).

Gradient

A vector operator that converts a scalar field into a vector field, pointing in the direction of the greatest rate of increase of the scalar field.

Divergence

A scalar operator that measures the outward flux density of a vector field at a point, indicating sources or sinks.

Curl

A vector operator that measures the rotational tendency or circulation of a vector field at a point.

Solenoidal Field

A vector field whose divergence is zero (∇ ⋅ F = 0), implying no sources or sinks within the field.

Irrotational Field

A vector field whose curl is zero (∇ × F = 0), implying no rotational tendency; such a field is conservative.

Conservative Field

An irrotational vector field for which the line integral between two points is independent of the path taken.

Vector Magnitude

Magnitude of a vector A = A_x i + A_y j + A_z k

|A| = \sqrt{A_x^2 + A_y^2 + A_z^2}

Unit Vector

Vector A divided by its magnitude gives a unit vector in the same direction.

\hat{u}_A = \frac{A}{|A|}

Dot Product (Scalar Product)

Yields a scalar. Used for projection, work done. θ is the angle between A and B.

A \· B = |A||B|\cos\θ = A_x B_x + A_y B_y + A_z B_z

Cross Product (Vector Product)

Yields a vector perpendicular to both A and B. Direction by right-hand rule. Used for torque, area of parallelogram.

A \× B = |A||B|\sin\θ \hat{n} = \begin{vmatrix} i & j & k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}

Scalar Triple Product

Volume of parallelepiped formed by vectors A, B, C.

A \· (B \× C) = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}

Vector Triple Product

Result is a vector in the plane of B and C.

A \× (B \× C) = (A \· C)B - (A \· B)C

Gradient of a Scalar Field

A vector field indicating the direction and magnitude of the maximum rate of increase of scalar field φ.

\nabla\φ = \frac{\partial\φ}{\partial x}i + \frac{\partial\φ}{\partial y}j + \frac{\partial\φ}{\partial z}k

Divergence of a Vector Field

A scalar field representing the outward flux density of vector field F at a point. Measures source/sink strength.

\nabla \· F = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Curl of a Vector Field

A vector field representing the rotational tendency or circulation of vector field F at a point.

\nabla \× F = \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}

Laplacian of a Scalar Field

Divergence of the gradient of a scalar field.

\nabla²\φ = \nabla \· (\nabla\φ) = \frac{\partial²\φ}{\partial x²} + \frac{\partial²\φ}{\partial y²} + \frac{\partial²\φ}{\partial z²}

Green's Theorem (in the plane)

Relates a line integral around a simple closed curve C to a double integral over the plane region R bounded by C.

\oint_C (P dx + Q dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA

Stokes' Theorem

Relates a line integral of a vector field F around a closed curve C to the surface integral of the curl of F over any surface S bounded by C.

\oint_C F \· dr = \iint_S (\nabla \× F) \· dS

Gauss' Divergence Theorem

Relates the surface integral of a vector field F over a closed surface S to the volume integral of the divergence of F over the volume V enclosed by S.

\iint_S F \· dS = \iiint_V (\nabla \· F) dV