- 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: \( \int_0^1 \int_0^2 (x+y) dy dx = 3 \).