Vector Analysis: Divergence and Curl

Flashcards covering key definitions and theorems related to divergence and curl of vector fields, including their expressions in different coordinate systems and the Divergence Theorem.

Divergence (div A)

In Cartesian coordinates, it is given by (∂Ax/∂x) + (∂Ay/∂y) + (∂Az/∂z), representing the net outward flux per unit volume at a point.

Del operator (∇)

An operator in Cartesian coordinates defined as (∂/∂x)âx + (∂/∂y)ây + (∂/∂z)âz, used in vector calculus operations like divergence and curl.

∇ • A

An alternative notation for div A, representing the divergence of vector A as the dot product of the del operator and vector A.

Divergence in Cylindrical Coordinates

The expression for ∇•A in cylindrical coordinates: (1/ρ)[∂(ρAρ)/∂ρ] + (1/ρ)(∂Aφ/∂φ) + (∂Az/∂z).

Divergence in Spherical Coordinates

The expression for ∇•A in spherical coordinates: (1/(r²sinθ))[∂(r²Ar)/∂r] + (1/(rsinθ))[∂(Aθsinθ)/∂θ] + (1/(rsinθ))(∂Aφ/∂φ).

Divergence Theorem

A theorem stating that the volume integral of the divergence of a vector field A equals the closed surface integral of A dot ds over the surface bounding that volume: ∫V (∇ • A)dv = ∮S A • ds.

Curl of a Vector Field (curl A or ∇ × A)

An indication of the tendency of a vector field A to 'push' or 'pull' along a closed path that encircles a point, representing the circulation per unit area.

Circulation

The tendency of a vector field to 'push' or 'pull' around a closed path, quantified by the line integral ∫ A • dℓ.

Right-Hand Rule (for Curl)

A rule determining the positive direction for the surface normal (ân) relative to the direction of a closed contour (Cj) bounding that surface; if fingers curl along Cj, the thumb points in the direction of ân.

Curl in Cartesian Coordinates

The expression for ∇ × A in Cartesian coordinates: (\frac{{\partial Az}}{{\partial y}} - \frac{{\partial Ay}}{{\partial z}})\hat{a}x + (\frac{{\partial Ax}}{{\partial z}} - \frac{{\partial Az}}{{\partial x}})\hat{a}y + (\frac{{\partial Ay}}{{\partial x}} - \frac{{\partial Ax}}{{\partial y}})\hat{a}z or in determinant form: |\begin{array}{ccc} \hat{a}x & \hat{a}y & \hat{a}z \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ Ax & Ay & A_z \end{array}|

Curl in Cylindrical Coordinates

The expression for ∇ × A in cylindrical coordinates: ( \frac{1}{\rho} \frac{{\partial Az}}{{\partial \phi}} - \frac{{\partial A\phi}}{{\partial z}} ) \hat{a}\rho + ( \frac{{\partial A\rho}}{{\partial z}} - \frac{{\partial Az}}{{\partial \rho}} ) \hat{a}\phi + ( \frac{1}{\rho} \frac{{\partial (\rho A\phi)}}{{\partial \rho}} - \frac{1}{\rho} \frac{{\partial A\rho}}{{\partial \phi}} ) \hat{a}_z

Curl in Spherical Coordinates

The expression for ∇ × A in spherical coordinates: \frac{1}{r \sin\theta} ( \frac{{\partial (A\phi \sin\theta)}}{{\partial \theta}} - \frac{{\partial A\theta}}{{\partial \phi}} ) \hat{a}r + \frac{1}{r} ( \frac{1}{\sin\theta} \frac{{\partial Ar}}{{\partial \phi}} - \frac{{\partial (r A\phi)}}{{\partial r}} ) \hat{a}\theta + \frac{1}{r} ( \frac{{\partial (r A\theta)}}{{\partial r}} - \frac{{\partial Ar}}{{\partial \theta}} ) \hat{a}_\phi

Physical Meaning of Curl

The curl of a vector field describes its 'rotation' or 'circulation' at a point. If curl A is non-zero, it indicates that the field has a tendency to rotate an object placed in it, or that there is a swirl in the field lines.

Stokes' Theorem

A theorem relating the surface integral of the curl of a vector field to the line integral of the vector field around the boundary of the surface: \ointC \mathbf{A} \cdot d\mathbf{l} = \iintS (\nabla \times \mathbf{A}) \cdot d\mathbf{S} where S is an open surface bounded by a closed contour C.