Unit 3: Multivariable Differential Calculus: Limits, Partials & Tangent Planes

1. Functions of Several Variables and Level Sets

A real-valued function of two variables is a rule $f: D \subseteq \mathbb{R}^2 \to \mathbb{R}$ that assigns to each ordered pair $(x, y)$ in a domain $D$ a unique real number $z = f(x, y)$.

• Graph: The set of all points $(x, y, z) \in \mathbb{R}^3$ such that $z = f(x, y)$ and $(x, y) \in D$. The graph forms a two-dimensional surface in $\mathbb{R}^3$.
• Level Curves (Contour Lines): The curves in $\mathbb{R}^2$ with equation $f(x, y) = c$, where $c$ is a constant in the range of $f$. A collection of level curves forms a contour map. Closely spaced contours indicate steep elevation changes; widely spaced contours indicate gentle slopes.
• Level Surfaces: For a function of three variables $w = F(x, y, z)$, the locus $F(x, y, z) = c$ defines a surface in $\mathbb{R}^3$.

2. Rigorous Definition of Multivariable Limits

Let $f$ be a function of two variables whose domain $D$ contains points arbitrarily close to $(a, b)$. We write:

if for every $\epsilon > 0$, there exists a corresponding $\delta > 0$ such that:

Geometrically, this requires $f(x, y)$ to be within $\epsilon$ of $L$ for all points $(x, y)$ inside an open punctured disk of radius $\delta$ centered at $(a, b)$, regardless of the path of approach.

3. The Two-Path Test for Non-Existence of Limits

In single-variable calculus, $x$ can approach $a$ from only two directions (left and right). In the plane, $(x, y)$ can approach $(a, b)$ along infinitely many distinct directions and curves (straight lines, parabolas, cubics, spirals).

Theorem (The Two-Path Test): If $f(x, y) \to L_1$ as $(x, y) \to (a, b)$ along a path $C_1$, and $f(x, y) \to L_2$ as $(x, y) \to (a, b)$ along a path $C_2$, where $L_1 \neq L_2$, then:

Critical Note on Parabolic Paths: It is not sufficient to check only straight lines $y = mx$. A function can approach the same value along every straight line through the origin, yet fail to have a limit along a parabolic trajectory such as $x = k y^2$.

4. Polar Coordinates Technique for Evaluating Limits

To prove that a limit $\lim_{(x, y) \to (0, 0)} f(x, y)$ equals $L$, convert to polar coordinates $x = r\cos\theta, y = r\sin\theta$:

If the resulting expression can be bounded by a function $g(r)$ that is independent of $\theta$ such that $\lim_{r \to 0^+} g(r) = 0$, then by the Squeeze Theorem the limit exists and equals $L$.

1. Definition and Geometric Interpretation of Partial Derivatives

Let $z = f(x, y)$. The partial derivative of $f$ with respect to $x$ at $(x_0, y_0)$ is the ordinary derivative of $f$ holding $y$ fixed at $y_0$:

Similarly, the partial derivative with respect to $y$ holds $x$ fixed at $x_0$:

Geometric Interpretation:

• The vertical plane $y = y_0$ intersects the surface $z = f(x, y)$ in a space curve $C_1$. The value $f_x(x_0, y_0)$ is the slope of the tangent line to $C_1$ at $(x_0, y_0, z_0)$.
• The vertical plane $x = x_0$ intersects the surface in a curve $C_2$. The value $f_y(x_0, y_0)$ is the slope of the tangent line to $C_2$ at $(x_0, y_0, z_0)$.

2. Higher-Order Partial Derivatives

For a function of two variables, there are four second-order partial derivatives:

3. Clairaut's Theorem (Equality of Mixed Partials)

Theorem (Alexis Clairaut / Hermann Schwarz): Let $f$ be defined on an open disk $D$ containing the point $(a, b)$. If the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are both continuous on $D$, then:

Proof Outline via the Mean Value Theorem:

Consider the difference quotient over a rectangle $[a, a+h] \times [b, b+k]$:

Define $g(x) = f(x, b+k) - f(x, b)$. Then $\Delta(h, k) = g(a+h) - g(a)$. By the single-variable Mean Value Theorem applied to $g$, there exists $\xi \in (a, a+h)$ such that:

Applying the Mean Value Theorem again to the function $y \mapsto f_x(\xi, y)$ on $[b, b+k]$, there exists $\eta \in (b, b+k)$ such that:

Similarly, defining $w(y) = f(a+h, y) - f(a, y)$ and applying the Mean Value Theorem in reverse order yields:

for some $\xi^ \in (a, a+h)$ and $\eta^ \in (b, b+k)$. Equating both expressions:

Taking the limit as $(h, k) \to (0, 0)$, by continuity of $f_{xy}$ and $f_{yx}$, $(\xi, \eta) \to (a, b)$ and $(\xi^, \eta^) \to (a, b)$:

1. Differentiability of Functions of Several Variables

In single-variable calculus, differentiability simply means the derivative $f'(x)$ exists. In multivariable calculus, the existence of both partial derivatives $f_x$ and $f_y$ is not sufficient to guarantee differentiability! (A surface can have cross-shaped tangent lines while being discontinuous elsewhere).

Definition (Differentiability in $\mathbb{R}^2$): A function $z = f(x, y)$ is differentiable at $(a, b)$ if the increment $\Delta z = f(a + \Delta x, b + \Delta y) - f(a, b)$ can be expressed in the form:

where $\epsilon_1, \epsilon_2 \to 0$ as $(\Delta x, \Delta y) \to (0, 0)$.

Sufficient Condition for Differentiability: If the partial derivatives $f_x$ and $f_y$ exist near $(a, b)$ and are continuous at $(a, b)$, then $f$ is guaranteed to be differentiable at $(a, b)$.

2. Tangent Plane to a Surface $z = f(x, y)$

Let $f(x, y)$ be differentiable at $(x_0, y_0)$, and let $z_0 = f(x_0, y_0)$.

The tangent line to the trace curve in $y = y_0$ has direction vector $\vec{v}_1 = \langle 1, 0, f_x(x_0, y_0) \rangle$. The tangent line to the trace curve in $x = x_0$ has direction vector $\vec{v}_2 = \langle 0, 1, f_y(x_0, y_0) \rangle$.

The normal vector to the tangent plane is the cross product:

The equation of the tangent plane at $(x_0, y_0, z_0)$ is:

3. Linearization and the Total Differential

• Linear Approximation (Linearization): The function $L(x, y)$ whose graph is the tangent plane is called the linearization of $f$ at $(a, b)$:

For points $(x, y)$ near $(a, b)$, $f(x, y) \approx L(x, y)$.

• The Total Differential:

Let $dx = \Delta x$ and $dy = \Delta y$ be independent increments. The total differential $dz$ is:

While $\Delta z$ represents the true change in $z$, $dz$ represents the change in height of the tangent plane.