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Vector Calculus

  • Understand the concept of vector functions and their derivatives.
  • Learn about partial derivatives and the gradient of a function.
  • Compute divergence and curl of a vector field.
  • Evaluate multiple integrals (double and triple integrals).
  • Apply line and surface integrals.
  • Understand the fundamental theorems of vector calculus.

Definition of Vector Functions

A **vector function** assigns a vector to each point in space. It is written as:

\[ \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k} \]

where \( x(t), y(t), z(t) \) are functions of \( t \).

Partial Derivatives and Gradient

If \( f(x,y) \) is a function of multiple variables, its **partial derivatives** are:

\[ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h} \] \[ \frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x, y+h) - f(x, y)}{h} \]

The gradient vector of \( f(x, y, z) \) is:

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

Divergence and Curl

The **divergence** of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is:

\[ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]

The **curl** of a vector field is:

\[ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} \]

Multiple Integrals: Double and Triple Integrals

A **double integral** over a region \( R \) is:

\[ \iint_R f(x,y) \, dA \]

A **triple integral** over a volume \( V \) is:

\[ \iiint_V f(x,y,z) \, dV \]

These are used to compute areas, volumes, and mass distributions.

Line and Surface Integrals

A **line integral** along a curve \( C \) for a scalar function is:

\[ \int_C f(x,y,z) \, ds \]

A **surface integral** over a surface \( S \) is:

\[ \iint_S F \cdot dS \]

Fundamental Theorems of Vector Calculus

  • Green’s Theorem: Relates a line integral around a simple curve to a double integral over the region inside.
    • \[ \oint_C (P dx + Q dy) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]
  • Stokes’ Theorem: Relates a surface integral to a line integral around its boundary.
    • \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \]
  • Gauss’ (Divergence) Theorem: Relates a flux integral over a closed surface to a volume integral.
    • \[ \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) dV \]

Examples

Example 1: Compute the gradient of \( f(x,y) = x^2 + y^3 \).

\[ \nabla f = (2x) \mathbf{i} + (3y^2) \mathbf{j} \]

Example 2: Evaluate the line integral \( \int_C (x^2 + y^2) \, ds \) where \( C \) is the unit circle \( x^2 + y^2 = 1 \).

Exercises

  • Question 1: Compute \( \nabla f \) for \( f(x,y,z) = x^2y + yz \).
  • Question 2: Find \( \nabla \cdot \mathbf{F} \) for \( \mathbf{F} = (xy, yz, zx) \).
  • Question 3: Evaluate the double integral \( \iint_R (x+y) \, dA \) over \( R: 0 \leq x \leq 1, 0 \leq y \leq 2 \).

  • Answer 1: \( \nabla f = (2xy) \mathbf{i} + (x^2 + z) \mathbf{j} + (y) \mathbf{k} \).
  • Answer 2: \( \nabla \cdot \mathbf{F} = y + z + x \).
  • Answer 3: \( \int_0^1 \int_0^2 (x+y) dy dx = 3 \).

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