Curl and Divergence Operators

Curl and Divergence

• Curl and Divergence Operators
  • Definition: Operators that act on vector fields.
  • Functionality: Produce a vector field (curl) and a scalar field (divergence) respectively.
  • Curl: Measures the rotation or "circulation" of a vector field at a given point. If a small paddle wheel were placed in the field, the curl would indicate how and where it would rotate.
  • Divergence: Measures the magnitude of a vector field's source or sink at a given point. If divergence is positive, it's a source; if negative, it's a sink; if zero, the field is incompressible (no net flow in or out).
• Vectors and Components
  • Given a vector field represented as:
    $\textbf{F} = P \textbf{i} + Q \textbf{j} + R \textbf{k}$
    where the partial derivatives of P, Q, R exist.
• Curl Definition: The curl of \textbf{F} is a vector field defined by:
  $\text{curl} \textbf{F} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{d}}{\text{d}x} & \frac{\text{d}}{\text{d}y} & \frac{\text{d}}{\text{d}z} \ P & Q & R \end{vmatrix} = \textbf{i} (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}) - \textbf{j} (\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) + \textbf{k} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})$
• Example Problem
  • If the vector field is defined as:
    $\textbf{F} = x^2 \textbf{i} + xyz \textbf{j} + y^2 \textbf{k}$
    Find curl:
  • The calculation is as follows:
    $\text{curl} \textbf{F} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{d}}{\text{d}x} & \frac{\text{d}}{\text{d}y} & \frac{\text{d}}{\text{d}z} \ x^2 & xyz & y^2 \end{vmatrix}$
• Theorem 1:
  • Statement: If f is a scalar field with continuous and ordered partial derivatives, the curl of the gradient of f is given by:
    $\text{curl} (\nabla f) = 0$
  • Proof Steps:
  • Expand the gradient:
    $\text{curl} (\nabla f) = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} \end{vmatrix}$
  • Confirmations: The second-order partial derivatives are commutative ($f<em>{xy} = f</em>{yx}$).
• Curl of a Conservative Vector Field:
  • Conclusion: The curl of any conservative vector field is zero.
• Example Problem - Non-Conservative Field:
  • Show that the vector field
    $\textbf{F} = -xz \textbf{i} + xyz \textbf{j} - y^2 \textbf{k}$
    is not conservative.
• Theorem 2:
  • If \textbf{F} is a vector field defined in an open region in $\textbf{R}^3$ that is simply connected, and curl \textbf{F} equals zero, then \textbf{F} is a conservative vector field.
• Example 3:
  • Analyze the vector field
    $\textbf{F}(x,y,z) = y^2z^3 \textbf{i} + 2xy^3z^3 \textbf{j} + 3x^2y^2z \textbf{k}$
    It is determined to be a conservative vector field in $\textbf{R}^3$ .
• Finding Potential Functions:
  • Find a scalar function such that
    $\textbf{F} = \nabla f$
  • Steps: Set the partial derivatives equal to the components of \textbf{F} and solve.
• Divergence:
  • Definition: The divergence of \textbf{F} is a scalar field defined as:
    $\text{div} \textbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$
• Theorem B:
  • Statement: If \textbf{F} is a vector field on an open region in $\textbf{R}^3$ with continuous partial derivatives, then:
    $\text{div(curl \textbf{F})} = 0$
• Vector Forms of Green's Theorem:
  • Vector field form is defined as:
    $\textbf{F} = P \textbf{i} + Q \textbf{j}$
  • Green's Theorem relates line integrals around a simple closed curve C to double integrals over the region D it encloses.
  • Tangent Form (Circulation Form): Relates the line integral of the tangential component of a vector field to the double integral of the curl's out-of-plane component.
    $\oint<em>C \textbf{F} \cdot d\textbf{r} = \iint</em>D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA = \iint_D (\text{curl} \textbf{F}) \cdot \textbf{k} dA$
  • Normal Form (Flux Form): Relates the line integral of the normal component of a vector field to the double integral of the divergence of the vector field.
    $\oint<em>C \textbf{F} \cdot \textbf{n} ds = \iint</em>D (\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}) dA = \iint_D \text{div} \textbf{F} dA$