Multivariable Calculus: Course Structure and Core Concepts (Notes)

Homework, Labs, and Course Structure

  • There are two separate homework submissions on the same date: online homework and written homework.

    • Written homework is on paper, needs to be scanned, converted to a PDF, and uploaded.
    • The submission system for the scanned homework is GradeScope.

  • Lab sessions are on Tuesdays (mornings and afternoons).

    • Students are divided into four groups per lab, with one GSI (registered instructor).
    • Each lab class has about 20–30 students and students work in groups of four.
    • Desmos (a free online graphics program) will be used; the instructor will demonstrate contents with Desmos.
    • Worksheets are prepared for the labs; the aim is to practice geometry in three dimensions.
    • The lab component includes a short worksheet at the end of each lab, which counts for 10% of the grade.

  • Course structure: three exams in total – two midterms and a final.

    • Students should check their calendars for dates; registrar dates are fixed and cannot be moved.
    • For serious conflicts, there is a web form (Google Form) to explain the conflict; course coordinator authorization is required; an alternate exam exists, but it requires a serious reason documented.

  • Gateway exam (open early in the semester): WebWork-based, available in September.

    • It covers Calculus I and II topics to ensure proficiency in one-variable techniques before moving to multivariable topics.
    • To pass the gateway, a student typically needs to score at least 7 out of 10.
    • The gateway is open for practice with multiple attempts; it begins next Monday/Tuesday and has a specific access route (basement, with directions). The gateway is designed to refresh one-variable skills before addressing multivariable problems.

  • AI use and integrity policies for homework

    • It is officially not allowed to rely on AI to solve homework problems, though AI tools exist and can be tempting.
    • The instructor uses an analogy (hiring someone to work out in a gym) to illustrate that the score should come from genuine effort, not outsourcing the work.
    • Some problems are graded by completion and some by more detailed solution (macros); there is no time to grade everything by macros.

  • Assignments and late submission policy

    • There are 12 assignments in total.
    • The last assignment will count for everyone; among the first 11 assignments, the two lowest scores are dropped (no questions asked).
    • Late homework is not accepted; the web work closes on a fixed date/time.
    • In the case of a major life event, exceptions can be considered, but generally late submissions are not allowed.

  • General classroom etiquette and study advice

    • Silence cell phones and show respect for the instructor and classmates.
    • The instructor emphasizes the need to stay engaged with the material and to keep a running list of concepts to avoid getting lost in a dense topic.
    • Form study groups and encourage collaboration; explain material to others or to yourself as a learning strategy.
    • Office hours and a math help room (math lab) are available frequently.
    • The key message: don’t just read math; actively engage with it.

  • Preview of course content and motivation for multivariable calculus

    • The instructor will introduce why three variables are needed and how functions of two variables look.
    • BMI example as a motivating application of multivariable functions:
    • BMI is a measure used by health professionals to assess healthy weight.
    • The BMI table presented shows height (in feet and inches) and weight (in pounds) as inputs and BMI as the output.
    • Example: with height 5'10" and weight 160 pounds, BMI = 23.
    • In population terms, height and weight are two independent variables that feed into a function to produce BMI.
    • In a two-variable function, inputs live on the xy-plane (height and weight correspond to a point (h, w)); the output (BMI) is a third coordinate (z) in space.

  • Graphs of multivariable functions and why 3D visualization is needed

    • For a function of two variables f(x,y), the inputs are pairs (x,y) on a region in the xy-plane; the output is f(x,y), which is plotted along the z-axis.
    • The graph of a two-variable function is a surface in three-dimensional space (R^3).
    • If you include a third independent variable (e.g., age), a three-variable function f(x,y,z) would, in principle, require a graph in four dimensions (R^4), which cannot be drawn directly in our three-dimensional intuition.
    • Therefore we often reason about higher-dimensional functions via projections and cross-sections rather than full graphs.

  • Three-dimensional coordinate geometry and Desmos visualization

    • Introduction to three mutually orthogonal coordinate axes (x, y, z) intersecting at the origin.
    • Any point P in space has coordinates (x,y,z) obtained by projecting onto the xy-plane and measuring height along z.
    • The xy-plane is defined by z = 0; the xz-plane by y = 0; the yz-plane by x = 0.
    • The eight octants are determined by the signs of (x,y,z); the first octant is where all three coordinates are nonnegative.
    • Desmos can be used to plot these planes (e.g., z = 0, x = 0, y = 0) and visualize the four quadrants in the xy-plane extended to 3D; the software provides a sense of the three-dimensional structure.

  • Distance formulas and their geometric interpretation

    • 2D distance: for points a = (a1, a2) and b = (b1, b2), the distance is

    d=(b<em>1−a</em>1)2+(b<em>2−a</em>2)2d = \sqrt{(b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2}

    • The base side length is |b1 − a1| and the height is |b2 − a2|; squaring and summing these components via Pythagoras yields the distance.
    • 3D distance: for points a = (a1, a2, a3) and b = (b1, b2, b3), the distance is

    d=(b<em>1−a</em>1)2+(b<em>2−a</em>2)2+(b<em>3−a</em>3)2d = \sqrt{(b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2 + (b<em>3 - a</em>3)^2}

    • This extends the 2D formula by adding the third coordinate difference and its square.

  • Sphere as a level surface of a distance function

    • The distance from a variable point (x,y,z) to a fixed center (1,2,3) is

    (x−1)2+(y−2)2+(z−3)2\sqrt{(x-1)^2 + (y-2)^2 + (z-3)^2}

    • A sphere of radius 5 centered at (1,2,3) is described by

    (x−1)2+(y−2)2+(z−3)2=25(x-1)^2 + (y-2)^2 + (z-3)^2 = 25

    • Points on the sphere satisfy that the distance to the center equals the radius; equivalently, the squared distance equals the radius squared.

  • A preview of vectors (to be covered next)

    • Distinguishes between static points and dynamic vectors; vectors will be introduced as objects with magnitude and direction, enabling description of motion and geometry in space.

  • Practical connections and deeper insight

    • The BMI example illustrates how a function of several variables can model a real-world quantity and how the graph concept generalizes beyond a single axis.
    • The discussion of dimensions clarifies why, for three input variables, we cannot simply draw the full graph in ordinary 3D space; the theoretical framework still guides reasoning about rates of change and level surfaces.

  • Quick conceptual recap of key ideas (ready for exam prep)

    • Functions with multiple inputs live in higher-dimensional domains: f: \mathbb{R}^n \to \mathbb{R}.
    • For n = 2, the graph is a surface in \mathbb{R}^3.
    • For n = 3, the graph would live in \mathbb{R}^4 (not directly drawable in 3D).
    • Distances in 2D and 3D follow the generalized Pythagorean theorem:

    d<em>2D=(b</em>1−a<em>1)2+(b</em>2−a<em>2)2,d</em>3D=(b<em>1−a</em>1)2+(b<em>2−a</em>2)2+(b<em>3−a</em>3)2d<em>{2D} = \sqrt{(b</em>1 - a<em>1)^2 + (b</em>2 - a<em>2)^2}, \quad d</em>{3D} = \sqrt{(b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2 + (b<em>3 - a</em>3)^2}

    • The equation of a sphere with center (h,k,l) and radius r is

    (x−h)2+(y−k)2+(z−l)2=r2(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2

  • Important terminology and sections to study next

    • 12.1 Coordinate Axes and three-dimensional coordinates; origin; projections; signs of coordinates; octants.
    • Concept of planes: x = 0, y = 0, z = 0 and planes like z = 0, to visualize coordinate geometry in 3D.
    • The downstream topic: vectors and their geometric interpretation to model directions and magnitudes in space.

Key formulas and quick reference

  • Distance in 2D:
    d=(b<em>1−a</em>1)2+(b<em>2−a</em>2)2d = \sqrt{(b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2}
  • Distance in 3D:
    d=(b<em>1−a</em>1)2+(b<em>2−a</em>2)2+(b<em>3−a</em>3)2d = \sqrt{(b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2 + (b<em>3 - a</em>3)^2}
  • Sphere with center $(h,k,l)$ and radius $r$:
    (x−h)2+(y−k)2+(z−l)2=r2(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2
    or equivalently for a fixed center, the locus of points is where the distance to the center equals $r$:
    (x−h)2+(y−k)2+(z−l)2=r\sqrt{(x-h)^2 + (y-k)^2 + (z-l)^2} = r
  • Graph of a two-variable function is a surface in 3D:
    Graph(f:R2→R)⊂R3\text{Graph}(f: \mathbb{R}^2 \to \mathbb{R}) \subset \mathbb{R}^3
  • Graph of a three-variable function would live in 4D:
    f:R3→R⇒conceptually in R4f: \mathbb{R}^3 \to \mathbb{R} \quad\Rightarrow\quad \text{conceptually in } \mathbb{R}^4
  • BMI as a function of height and weight (two independent variables):
    • Inputs: height $H$ and weight $W$; output BMI.
    • Represented as a function: BMI:R2→RBMI: \mathbb{R}^2 \to \mathbb{R}
    • Example given: height 5'10" and weight 160 lb yields BMI = 23.
  • Coordinate geometry basics in 3D
    • Origin is the intersection point of the three axes (0,0,0).
    • The x-axis, y-axis, z-axis extend in perpendicular directions from the origin.
    • The four planes and eight octants arise from the signs of $x$, $y$, and $z$; first octant: $x \ge 0$, $y \ge 0$, $z \ge 0$.
  • Desmos and 3D visualization notes
    • Desmos supports visualizations like planes $z=0$, $x=0$, $y=0$ and can rotate and inspect 3D-like representations to aid intuition.

Connections to foundational principles

  • The transition from 1-variable calculus to multivariable calculus involves moving from graphs in 2D (curves) to graphs in higher dimensions (surfaces, hypersurfaces).
  • Independence of variables (e.g., height and weight) leads to functions of multiple inputs and the need for extra axes to represent outputs.
  • The concept of level sets, surfaces, and three-dimensional geometry plays a central role in understanding optimization, differential geometry, and applied problems (e.g., BMI example).
  • The use of visual tools (Desmos, computer labs) is emphasized because much of multivariable calculus involves geometry of surfaces in 3D space that is hard to visualize on paper alone.

Worked example scaffolding (conceptual, not exhaustive)

  • Given two inputs $(a1,a2)$ and $(b1,b2)$, compute the 2D distance using the formula above.
  • For a box with opposite vertices $(a1,a2,a3)$ and $(b1,b2,b3)$, compute the space diagonal using
    d=(b<em>1−a</em>1)2+(b<em>2−a</em>2)2+(b<em>3−a</em>3)2d = \sqrt{(b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2 + (b<em>3 - a</em>3)^2}
  • Interpret a sphere as the set of points at a fixed distance from its center; this connects the distance formula to a familiar geometric surface.

End of notes excerpt