Vector calculus

Vector Fields

Definition

A vector field $\mathbf{F}$ in 3D space assigns a vector $\mathbf{F}(x,y,z)$ to each point $(x,y,z)$ in its domain: \(\mathbf{F}(x,y,z) = P(x,y,z)\mathbf{i} + Q(x,y,z)\mathbf{j} + R(x,y,z)\mathbf{k}\)

In 2D space, a vector field has the form: \(\mathbf{F}(x,y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}\)

Conservative Vector Fields

A vector field $\mathbf{F}$ is conservative if there exists a scalar function $f$ such that $\mathbf{F} = \nabla f$. In this case:

• $f$ is called the potential function of $\mathbf{F}$
• Line integrals of $\mathbf{F}$ are path-independent
• $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ for any closed curve $C$

Criteria for Conservative Fields

A vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is conservative if and only if:

• $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$
• $\frac{\partial P}{\partial z} = \frac{\partial R}{\partial x}$
• $\frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}$

In a simply connected domain, these conditions are equivalent to $\nabla \times \mathbf{F} = \mathbf{0}$.

Differential Operators

Gradient

The gradient of a scalar function $f(x,y,z)$ is a vector field: \(\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}\)

Properties:

• $\nabla f$ points in the direction of steepest increase of $f$
• $\nabla f$ is perpendicular to level surfaces of $f$

• The magnitude $

  \nabla f

  $ gives the rate of change in the direction of steepest increase

Divergence

The divergence of a vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is a scalar field: \(\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\)

Physical interpretation:

• Measures the rate at which “flux” exits a small region around a point
• Positive divergence: source (field lines flow outward)
• Negative divergence: sink (field lines flow inward)
• Zero divergence: no net creation or destruction (incompressible flow)

Curl

The curl of a vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is a vector field: \(\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}\)

Physical interpretation:

• Measures the rotational tendency of the field
• The direction of curl is the axis of rotation
• The magnitude of curl gives the strength of rotation
• Zero curl: irrotational field (no rotational component)

Laplacian

The Laplacian of a scalar function $f$ is: \(\nabla^2 f = \nabla \cdot (\nabla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\)

The Laplacian of a vector field $\mathbf{F}$ is: \(\nabla^2 \mathbf{F} = \nabla^2 P \mathbf{i} + \nabla^2 Q \mathbf{j} + \nabla^2 R \mathbf{k}\)

Applications:

• Appears in many physical equations (heat equation, wave equation, Poisson’s equation)
• $\nabla^2 f = 0$ defines harmonic functions (solutions to Laplace’s equation)

Vector Calculus Identities

Basic Identities

• $\nabla(f + g) = \nabla f + \nabla g$
• $\nabla(fg) = f\nabla g + g\nabla f$
• $\nabla \cdot (f\mathbf{F}) = f(\nabla \cdot \mathbf{F}) + \mathbf{F} \cdot \nabla f$
• $\nabla \times (f\mathbf{F}) = f(\nabla \times \mathbf{F}) + (\nabla f) \times \mathbf{F}$
• $\nabla \cdot (\mathbf{F} \times \mathbf{G}) = \mathbf{G} \cdot (\nabla \times \mathbf{F}) - \mathbf{F} \cdot (\nabla \times \mathbf{G})$

Second-Order Identities

• $\nabla \times (\nabla f) = \mathbf{0}$ (curl of a gradient is zero)
• $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ (divergence of a curl is zero)
• $\nabla \times (\nabla \times \mathbf{F}) = \nabla(\nabla \cdot \mathbf{F}) - \nabla^2 \mathbf{F}$

Worked Examples

Example 1: Determining if a Vector Field is Conservative

Determine if the vector field $\mathbf{F}(x,y,z) = (y^2+z) \mathbf{i} + (2xy-z) \mathbf{j} + (x-y) \mathbf{k}$ is conservative, and if so, find a potential function.

Solution:

1. Check if $\nabla \times \mathbf{F} = \mathbf{0}$:
   • $\frac{\partial}{\partial y}(x-y) - \frac{\partial}{\partial z}(2xy-z) = -1 - (-1) = 0$
   • $\frac{\partial}{\partial z}(y^2+z) - \frac{\partial}{\partial x}(x-y) = 1 - 1 = 0$
   • $\frac{\partial}{\partial x}(2xy-z) - \frac{\partial}{\partial y}(y^2+z) = 2y - 2y = 0$

   Since $\nabla \times \mathbf{F} = \mathbf{0}$, the field is conservative.

2. Find the potential function $f$ such that $\mathbf{F} = \nabla f$:
   • $\frac{\partial f}{\partial x} = y^2+z$
   • $\frac{\partial f}{\partial y} = 2xy-z$
   • $\frac{\partial f}{\partial z} = x-y$
3. Integrate $\frac{\partial f}{\partial x} = y^2+z$ with respect to $x$:
   • $f(x,y,z) = xy^2 + xz + g(y,z)$, where $g(y,z)$ is a function of $y$ and $z$ only
4. Differentiate with respect to $y$:
   • $\frac{\partial f}{\partial y} = 2xy + \frac{\partial g}{\partial y}$
   • Comparing with $\frac{\partial f}{\partial y} = 2xy-z$, we get:
   • $\frac{\partial g}{\partial y} = -z$
5. Integrate with respect to $y$:
   • $g(y,z) = -yz + h(z)$, where $h(z)$ is a function of $z$ only
6. Differentiate with respect to $z$:
   • $\frac{\partial f}{\partial z} = x - y + \frac{dh}{dz}$
   • Comparing with $\frac{\partial f}{\partial z} = x-y$, we get:
   • $\frac{dh}{dz} = 0$, so $h(z) = C$ (a constant)
7. Therefore, the potential function is:
   • $f(x,y,z) = xy^2 + xz - yz + C$

Strategies for Solving Vector Calculus Problems

For Testing if a Field is Conservative

1. Compute the curl: $\nabla \times \mathbf{F}$
2. Check if the curl is identically zero
3. If yes, find the potential function by integrating the components
4. Verify your potential function by taking its gradient

Connections to Linear Algebra

The concepts in vector calculus build directly on fundamental ideas from linear algebra:

• Vectors and vector operations from [[ linear_algebra/vector_geometry/Vectors ]]
• Dot products from Dot (Inner) Product
• Cross products from Cross Product
• Vector spaces from Vector Spaces

See

• Vector Geometry
• Gradients
• Multiple Integrals
• Derivatives - For foundation of differential operators
• Integration Techniques - For foundation of integral operators

  See

• Multivariable Calculus

• Vector Geometry (in Linear Algebra)

• Gradients

• Multiple Integrals