Calculus III Syllabus and Topical Outline: Multivariable Calculus and Vector Calculus

Course Overview

This course covers advanced topics in calculus, focusing on real-valued functions of several variables and vector field theory. Students will learn to analyze and compute limits, derivatives, and integrals in higher dimensions, as well as apply these concepts to geometric and physical problems.

Topical Outline

Vectors and the Geometry of Space

Vectors are fundamental objects in mathematics and physics, representing quantities with both magnitude and direction. Understanding their properties and operations is essential for multivariable calculus.

• Vector Operations: Includes addition, scalar multiplication, dot product, and cross product.

• Applications: Used to describe lines, planes, and geometric surfaces in three-dimensional space.

• Equations of Lines and Planes: Find equations using vector and parametric forms.

• Quadric Surfaces and Cylinders: Identify and classify surfaces such as ellipsoids, paraboloids, and hyperboloids.

Vector-Valued Functions and Space Curves

Vector-valued functions describe curves and motion in space. Calculus tools are used to analyze their properties.

• Limits, Derivatives, and Integrals: Compute these for vector-valued functions of one variable.

• Curve Analysis: Find arc length, curvature, tangent lines, unit tangent vectors, principal unit normal vectors, and binormal vectors.

Functions of Several Variables

Functions with more than one variable require new calculus techniques for analysis and optimization.

• Limits and Continuity: Evaluate limits and determine continuity for multivariable functions.

• Partial Derivatives: Compute derivatives with respect to each variable.

• Directional Derivatives and Gradients: Find rates of change in arbitrary directions and the gradient vector.

• Tangent Planes and Extrema: Use differentiation to find tangent planes, relative extrema, and absolute extrema on closed and bounded regions.

• Lagrange Multipliers: Apply this method to find extrema subject to constraints.

Multiple Integration

Integration in higher dimensions allows calculation of areas, volumes, and other physical quantities.

• Double and Triple Integrals: Evaluate integrals in two and three dimensions.

• Coordinate Systems: Use Cartesian, polar, cylindrical, and spherical coordinates for integration.

• Applications: Calculate areas, volumes, surface areas, mass, and centers of mass.

Vector Calculus

Vector calculus extends integration and differentiation to vector fields, with important applications in physics and engineering.

• Line Integrals: Evaluate scalar and work integrals along curves.

• Surface and Flux Integrals: Integrate over surfaces and compute flux through surfaces.

• Fundamental Theorems: Apply the Fundamental Theorem of Calculus for Line Integrals, Green's Theorem, Stokes' Theorem, and the Divergence Theorem.

• Conservative Vector Fields: Identify conservative fields and find potential functions.

Key Formulas and Concepts

• Dot Product: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$

• Cross Product: $\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)$

• Equation of a Plane: $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$

• Gradient: $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$

• Double Integral (Cartesian): $\iint_R f(x, y)\,dx\,dy$

• Double Integral (Polar): $\iint_R f(r, \theta)\,r\,dr\,d\theta$

• Triple Integral (Cylindrical): $\iiint_E f(r, \theta, z)\,r\,dr\,d\theta\,dz$

• Triple Integral (Spherical): $\iiint_E f(\rho, \phi, \theta)\,\rho^2 \sin\phi\,d\rho\,d\phi\,d\theta$

• Line Integral: $\int_C \vec{F} \cdot d\vec{r}$

• Green's Theorem: $\oint_C (M\,dx + N\,dy) = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx\,dy$

• Stokes' Theorem: $\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$

• Divergence Theorem: $\iint_S \vec{F} \cdot d\vec{S} = \iiint_E (\nabla \cdot \vec{F})\,dV$

Example Applications

• Finding the equation of a plane: Given a point $(1,2,3)$ and normal vector $(4,5,6)$, the equation is $4(x-1) + 5(y-2) + 6(z-3) = 0$.

• Computing a double integral in polar coordinates: To find the area inside a circle of radius $a$, use $\iint_R 1\,r\,dr\,d\theta = \int_0^{2\pi} \int_0^a r\,dr\,d\theta = \pi a^2$.

• Using Lagrange multipliers: To maximize $f(x, y)$ subject to $g(x, y) = 0$, solve $\nabla f = \lambda \nabla g$.

Table: Comparison of Coordinate Systems for Multiple Integration

Coordinate System	Variables	Integration Element	Typical Use
Cartesian	x, y, z	$dx\,dy\,dz$	Rectangular regions
Polar	r, $\theta$	$r\,dr\,d\theta$	Circular regions
Cylindrical	r, $\theta$, z	$r\,dr\,d\theta\,dz$	Cylindrical symmetry
Spherical	$\rho$, $\phi$, $\theta$	$\rho^2 \sin\phi\,d\rho\,d\phi\,d\theta$	Spherical symmetry

Additional info: These notes are based on the syllabus and topical outline for a Calculus III (Multivariable Calculus) course, covering vectors, multivariable functions, multiple integration, and vector calculus theorems.