QUIZ

Gradient

Definition / Explanation / Theory

The gradient of a scalar field \(\phi(x, y, z)\) is a vector field that points in the direction of the maximum rate of increase of \(\phi\).
It represents how fast and in which direction the scalar field changes in space.

Mathematically:

\[\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}\]

Important Formulas

  • Directional Derivative: \(\frac{d\phi}{ds} = \nabla \phi \cdot \hat{n}\) where \(\hat{n}\) is a unit vector in the direction of change.

  • Magnitude of Gradient: \(|\nabla \phi| = \sqrt{\left(\frac{\partial \phi}{\partial x}\right)^2 + \left(\frac{\partial \phi}{\partial y}\right)^2 + \left(\frac{\partial \phi}{\partial z}\right)^2}\)

Physical Interpretation

  • The gradient points toward the steepest ascent of \(\phi\).
  • In heat flow, \(-\nabla T\) gives the direction of heat flow.
  • In electrostatics, \(\mathbf{E} = -\nabla V\), where \(V\) is the electric potential.

Important Points

  • Gradient converts a scalar → vector field.
  • The gradient is perpendicular to the level surfaces (equipotential surfaces).
  • Zero gradient implies the scalar field is constant in that region.
  • Units of gradient = (units of \(\phi\)) / (units of length).

Divergence

Definition / Explanation / Theory

The divergence of a vector field \(\mathbf{A}(x, y, z)\) measures the net rate of flux expansion or contraction per unit volume at a point.
It indicates whether the vector field behaves like a source or a sink at that point.

Mathematically:

\[\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\]

Important Formulas

  • In Cartesian Coordinates: \(\nabla \cdot \mathbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\)

  • In Cylindrical Coordinates: \(\nabla \cdot \mathbf{A} = \frac{1}{r}\frac{\partial (rA_r)}{\partial r} + \frac{1}{r}\frac{\partial A_\theta}{\partial \theta} + \frac{\partial A_z}{\partial z}\)

  • In Spherical Coordinates: \(\nabla \cdot \mathbf{A} = \frac{1}{r^2}\frac{\partial (r^2 A_r)}{\partial r} + \frac{1}{r\sin\theta}\frac{\partial (A_\theta \sin\theta)}{\partial \theta} + \frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi}\)

Physical Interpretation

  • If \(\nabla \cdot \mathbf{A} > 0\): field diverges → source.
  • If \(\nabla \cdot \mathbf{A} < 0\): field converges → sink.
  • If \(\nabla \cdot \mathbf{A} = 0\): field is solenoidal (no net flux).

Examples

  • Fluid flow: \(\nabla \cdot \mathbf{v}\) gives rate of expansion of fluid.
  • Electrostatics: \(\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\) (Gauss’s law in differential form).
  • Magnetostatics: \(\nabla \cdot \mathbf{B} = 0\) (no magnetic monopoles).

Important Points

  • Divergence converts a vector → scalar field.
  • Units of divergence = (units of \(A\)) / (length).
  • Used in continuity equation, Gauss’s theorem, and field theory.

Curl

Definition / Explanation / Theory

The curl of a vector field \(\mathbf{A}(x, y, z)\) measures the tendency of rotation or circulation of the field around a point.
It represents the rotational nature of a vector field.

Mathematically:

\[\nabla \times \mathbf{A} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix} = \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right)\hat{i} + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right)\hat{j} + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right)\hat{k}\]

Important Formulas

  • Magnitude of Curl: \(|\nabla \times \mathbf{A}| = 2\omega\) where \(\omega\) is the angular velocity vector of rotation.

  • In Cylindrical Coordinates:

\[\nabla \times \mathbf{A} = \left( \frac{1}{r}\frac{\partial A_z}{\partial \theta} - \frac{\partial A_\theta}{\partial z} \right)\hat{r} + \left( \frac{\partial A_r}{\partial z} - \frac{\partial A_z}{\partial r} \right)\hat{\theta} + \frac{1}{r}\left( \frac{\partial (rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right)\hat{z}\]
  • In Spherical Coordinates: (Formula not required at this level but useful for field problems.)

Physical Interpretation

  • Curl measures local rotation in the vector field.
  • In fluid mechanics, \(\nabla \times \mathbf{v}\) gives vorticity.
  • In electromagnetism, \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) (Faraday’s law).

Important Points

  • Curl converts a vector → vector field.
  • If \(\nabla \times \mathbf{A} = 0\), field is irrotational.
  • Direction of curl vector gives axis of rotation, and magnitude gives strength of rotation.
  • Units of curl = (units of \(A\)) / (length).
  • Common in rotational motion, fluid mechanics, and Maxwell’s equations.