Calculus II - Course Notes

1. Introduction to Euclidean Space ($\R^m$)

1.1 Vectors and Norms

In an Euclidean space $\R^n$, vectors are represented as:

Operations:

• Sum of vectors: $\vecx + \vecy = (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n)$
• Scalar multiplication: $c \in \R \implies c\vecx = (cx_1, cx_2, \dots, cx_n)$
• Scalar (Dot) Product: $\vecx \cdot \vecy = x_1y_1 + x_2y_2 + \dots + x_ny_n$

Norm/Metric:

The norm assigns to each vector a non-negative number:

Distance between two points: $d(\vecx, \vecy) = \norm{\vecx - \vecy}$.

Properties of Norm:

1. $\norm{\vecx + \vecy} \le \norm{\vecx} + \norm{\vecy}$ (Triangle Inequality)
2. $\norm{\vecx} \ge 0$, and $\norm{\vecx} = 0 \iff \vecx = \vec{0}$

2. Topology of $\R^m$

2.1 Balls and Neighborhoods

• Open Ball: $B_r(\veca) = \{ \vecx \in \R^m : \norm{\vecx - \veca} < r \}$, where $r$ is the radius and $\veca$ is the center.
• In $\R^2$, the unit ball centered at the origin is $B_1((0,0)) = \{ (x,y) : \sqrt{x^2+y^2} < 1 \}$.

2.2 Interior, Boundary, and Closure

Given a domain $D \subset \R^m$:

• Interior Point: $\vec{p} \in D$ is an interior point if there exists $\epsilon > 0$ such that $B_\epsilon(\vec{p}) \subseteq D$.
• Boundary Point: $\vec{p}$ is a boundary point of $D$ if for every $\epsilon > 0$, the ball $B_\epsilon(\vec{p})$ contains points both inside $D$ and outside $D$ ($B_\epsilon \cap D \neq \emptyset$ and $B_\epsilon \cap (\R^m \setminus D) \neq \emptyset$).
• Isolated Point: $\vec{p} \in D$ is isolated if there exists $\epsilon > 0$ such that $B_\epsilon(\vec{p}) \cap D = \{\vec{p}\}$.
• Open Set: A set $D$ is open if all its points are interior points.
• Closed Set: A set $D$ is closed if its complement ($\R^m \setminus D$) is open, or equivalently, if it contains all its boundary points ($\partial D \subseteq D$).
• Closure: $\overline{D} = D \cup \partial D$.
• Accumulation Point: $\veca \in \R^m$ is an accumulation point if every $B_\epsilon(\veca)$ contains at least one point of $D$ other than $\veca$.

3. Multivariable Functions

A scalar function of several variables is a mapping $f: \R^m \rightarrow \R$.

3.1 Domains and Graphs

The graph of a function $f(x,y)$ is a surface in $\R^3$:

3.2 Level Curves and Surfaces

For a constant $c \in \R$, a level curve is the set:

Example: For $f(x,y) = \sqrt{9-x^2-y^2}$, setting $z=c$ yields circles $x^2+y^2 = 9-c^2$ for $0 \le c \le 3$.

4. Limits and Continuity

4.1 Definition of a Limit

Suppose $\veca$ is an accumulation point of the domain $D$. We say $\lim_{\vecx \to \veca} f(\vecx) = L$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that:

Theorem: If a limit exists, it is unique.

4.2 Evaluating Limits

To prove a limit does not exist, show that approaching $(0,0)$ along different paths (e.g., $x=0$, $y=0$, or $y=mx$) yields different results.

4.3 Continuity

$f: \R^m \rightarrow \R$ is continuous at $\veca \in D$ if:

Theorem: A continuous function on a closed and bounded set attains its absolute maximum and minimum values.

5. Derivatives

5.1 Partial Derivatives

For $f(x,y)$, the partial derivatives at $(a,b)$ are:

Clairaut's Theorem: If $f_{xy}$ and $f_{yx}$ are continuous, they are equal ($f_{xy} = f_{yx}$).

5.2 The Gradient and Hessian Matrix

The gradient is a vector field:

The Hessian is the symmetric matrix of second partial derivatives:

5.3 Differentiability and Tangent Planes

A function $f: \R^2 \rightarrow \R$ is differentiable at $\veca = (a,b)$ if there exists a linear operator $d_{\veca}f$ such that:

Tangent Plane Equation:

Theorem: If the partial derivatives exist and are continuous at $\veca$, then $f$ is differentiable at $\veca$. If $f$ is differentiable, then $f$ is continuous.

6. Chain Rule and Directional Derivatives

6.1 The Chain Rule

If $f = f(x,y,z)$ and $x = x(u,v), y = y(u,v), z = z(u,v)$, then:

6.2 Directional Derivatives

The rate of change of $f$ in the direction of a unit vector $\vec{u}$ is:

• The directional derivative is maximal in the direction of the gradient $\nabla f(\vecx_0)$.
• The maximal value of the rate of change is $\norm{\nabla f(\vecx_0)}$.
• The direction of minimum rate of change is $-\nabla f(\vecx_0)$.

7. Optimization

7.1 Local Maxima and Minima

A function $f$ has a local maximum at $\veca$ if $f(\vecx) \le f(\veca)$ for all $\vecx$ in some neighborhood $B_\epsilon(\veca)$.

Candidates for local extrema are critical points, where $\nabla f(\veca) = \vec{0}$ or $\nabla f$ does not exist.

7.2 Second Derivative Test

Let $\veca$ be a critical point ($\nabla f(\veca) = \vec{0}$) and define $\Delta_2 = \det(H) = f_{xx}f_{yy} - (f_{xy})^2$.

1. If $\Delta_2 > 0$ and $f_{xx} > 0$, $f$ has a local minimum at $\veca$.
2. If $\Delta_2 > 0$ and $f_{xx} < 0$, $f$ has a local maximum at $\veca$.
3. If $\Delta_2 < 0$, $f$ has a saddle point at $\veca$.
4. If $\Delta_2 = 0$, the test is inconclusive.

7.3 Absolute Maxima and Minima

To find the absolute maximum and minimum on a closed, bounded region $D$:

1. Find all critical points of $f$ inside $D$ and evaluate $f$ at these points.
2. Find the extreme values of $f$ on the boundary $\partial D$ (often using parameterization or Lagrange multipliers).
3. The largest of these values is the absolute maximum; the smallest is the absolute minimum.

8. Lagrange Multipliers

To maximize or minimize $f(\vecx)$ subject to a constraint $g(\vecx) = k$:

1. Set up the system of equations: $\nabla f = \lambda \nabla g$ and $g(\vecx) = k$.
2. Solve the system for $(x, y, z, \lambda)$.
3. Evaluate $f$ at all solution points to find the extrema.