Exhaustive Calculus III Study Guide: 3D Coordinate Systems, Vectors, and Course Logistics

Course Logistics, Grading, and Exam Policies

• Office Hours and Instructor Contact:

  • Office hours are scheduled on Wednesdays from 10:00 AM to 3:00 PM.

  • Office hours are open drop-in session; no prior appointment is necessary. Attendance can range from five minutes to the full block.

  • Office hours cover homework help, course material discussions, or general introductions.

  • Contact via email must be sent to the official tamu.edu email address rather than Canvas inbox messages, as Canvas messages get buried under unread notifications.

  • If requesting an appointment outside scheduled office hours, specific available days and time slots must be provided in the email.

• Canvas Course Structure & Exam Coverage:

  • Course modules are organized directly by exam coverage:

  • Exam 1: Chapter 12 and Chapter 13

  • Exam 2: Chapter 14

  • Exam 3: Chapter 15

  • Final Exam: Chapter 16

  • Unlike Calculus I and Calculus II, the Final Exam is not strictly comprehensive. However, Chapter 16 relies on foundational concepts learned throughout the entire semester.

• Online Homework Platform (Cengage WebAssign):

  • Homework is conducted through Cengage and accounts for $10\%$ of the final course grade.

  • Initial access to Cengage must be completed by clicking the direct module link in Canvas to populate the course roster.

  • Each question allows up to $4$ submission attempts.

  • Correctness is binary (right or wrong) regardless of whether it is solved on the first or fourth try.

  • Questions can be submitted part-by-part. Submitting partial answers only consumes attempts for the specific input fields filled; blank fields retain their remaining attempts.

  • Students are advised not to exhaust all $4$ attempts before seeking assistance, as technical/formatting issues with Cengage may occur, and additional tries cannot be granted by the instructor.

  • Freshmen receive an initial $10$-day free trial access window. After $10$ days, a paid subscription (such as Cengage Unlimited) is required.

• Textbook Information:

  • Textbook: Calculus: Early Transcendentals, 9th Edition by James Stewart.

  • Access to the e-book is included with the Cengage homework subscription.

  • The e-book features embedded instructional videos and examples at the end of each section.

  • The same textbook spans Calculus I, Calculus II, and Calculus III. Calculus II ends at Chapter 11 (Sequences and Series); Calculus III begins at Chapter 12.

  • Pre-requisite vector material from Calculus I is located in Appendix J of Stewart's book.

• Exams, Accommodations, and Attendance Policies:

  • Exams fit into standard $75$-minute lecture slots and contain both multiple-choice and workout sections.

  • All exams take place in the standard classroom unless accommodations are registered with the disability testing center.

  • Students testing at the accommodation center should reserve morning slots due to scheduling overlaps with university common exams.

  • Absence from graded assignments or exams requires a University Approved Excuse per Student Rule 7.

  • Departmental makeup exam dates are held on designated Fridays and Mondays.

Academic Support and Learning Resources

• Math Learning Center (MLC) Help Sessions:

  • Free walk-in tutoring sessions located in the Blocker Building.

  • Hours span Monday through Friday (e.g., Mondays operate from 10:00 AM to 7:00 PM for both in-person and online assistance).

• Week in Review (WIR):

  • Led by Professor Brooks.

  • Held live every Tuesday from 7:30 PM to 9:30 PM in Blocker 166, with a concurrent Zoom stream accessible via the MLC website.

  • Blank note templates are posted in advance; fully worked solutions are posted following the session.

• Virtual Math Learning Center (VMLC):

  • Online repository containing video lectures and supplemental practice problems created specifically by TAMU professors for TAMU students.

  • Requires login via NetID and password.

  • Includes review materials for prerequisite courses, such as Math 150 (Precalculus).

• Python Concept Animations:

  • Short, $5$-minute conceptual videos featuring Python animations and voiceovers.

  • Covers specific abstract topics such as vector equations of 3D lines, planes, and Lagrange multipliers (does not cover problem-solving procedures).

• Class Materials & Formats:

  • Lecture notes are provided on Google Drive in accessible Word document format as well as historical PDF files.

  • Coordinated practice exams with complete solution keys are released prior to major exams.

Three-Dimensional Coordinate Systems

• Defining $R^3$ Space:

  • $R^2$ (Two-Dimensional Space) consists of two perpendicular axes ($x$ and $y$) dividing the plane into $4$ quadrants.

  • $R^3$ (Three-Dimensional Space) consists of three mutually perpendicular coordinate axes intersecting at the origin $(0, 0, 0)$, dividing space into $8$ octants.

• Spatial Calibration and Orientations:

  • The front board represents the $yz$-plane.

  • The horizontal axis along the board is the $y$-axis.

  • The vertical axis along the board is the $z$-axis.

  • The $x$-axis sticks straight out perpendicular from the board into the room space.

  • The left boundary wall represents the $xz$-plane.

  • The floor represents the $xy$-plane.

  • The First Octant corresponds to the region where all coordinates are positive ($x > 0$, $y > 0$, $z > 0$).

• Right-Hand Thumb Rule:

  • Used to define the positive direction of the $z$-axis.

  • Curl the fingers of the right hand from the positive $x$-axis toward the positive $y$-axis.

  • The extended right thumb points in the direction of the positive $z$-axis.

• Locating Points in $R^3$:

  • Points are specified in ordered triplet form $P(x, y, z)$.

  • To locate $P(3, 5, 4)$, perform the following sequence:

  1. Move $3$ units forward along the positive $x$-axis.

  2. Move $5$ units right parallel to the positive $y$-axis.

  3. Move $4$ units upward parallel to the positive $z$-axis.

Projections and Geometric Representations in $R^3$

• Projections of Points onto Coordinate Planes:

  • A projection drops a point perpendicularly onto a given coordinate plane.

  • On the $yz$-plane, $x = 0$. Projection of $P(3, 5, 4)$ is $(0, 5, 4)$.

  • On the $xy$-plane, $z = 0$. Projection of $P(3, 5, 4)$ is $(3, 5, 0)$.

  • On the $xz$-plane, $y = 0$. Projection of $P(3, 5, 4)$ is $(3, 0, 4)$.

• Points Located on Coordinate Axes:

  • On the $x$-axis, $y = 0$ and $z = 0$ ($(x, 0, 0)$).

  • On the $y$-axis, $x = 0$ and $z = 0$ ($(0, y, 0)$).

  • On the $z$-axis, $x = 0$ and $y = 0$ ($(0, 0, z)$).

• Curves in $R^2$ versus Surfaces in $R^3$:

  • An equation with two variables represents a 1D curve in $R^2$, but represents a 2D surface extending along the missing variable in $R^3$

  • Example 1 ($y = x$):

  • In $R^2$: A straight 2D line passing through the origin at a $45^\circ$ angle.

  • In $R^3$: A vertical plane passing through $y = x$ in the $xy$-plane, stretching infinitely along the $z$-axis.

  • Example 2 ($y = x^2 + 1$):

  • In $R^2$: A standard parabola opening upward with vertex at $(0, 1)$.

  • In $R^3$: A parabolic cylinder extending along the $z$-axis.

Distance Formula and Equations of Spheres

• Distance Formula in $R^3$:

  • Derived by applying the Pythagorean theorem twice across three dimensional components.

  • Distance $d$ between $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$:\n  d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\n  

• Standard Equation of a Sphere:

  • A sphere is defined as the set of all points $P(x, y, z)$ at a fixed distance (radius $r$) from a central point $C(h, k, l)$:\n  (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\n  

• Finding Center and Radius by Completing the Square:

  • Problem: Find center and radius of $x^2 + y^2 + z^2 - 12x - 6y + 6z + 45 = 0$

  • Step 1: Group variables and move constant term to the right side:\n  (x^2 - 12x) + (y^2 - 6y) + (z^2 + 6z) = -45\n  

  • Step 2: Complete the square for each variable using $\left(\frac{b}{2}\right)^2$:

  • For $x$: $\left(\frac{-12}{2}\right)^2 = 36$

  • For $y$: $\left(\frac{-6}{2}\right)^2 = 9$

  • For $z$: $\left(\frac{6}{2}\right)^2 = 9$

  • Step 3: Balance equation by adding added terms to the right side:\n  (x^2 - 12x + 36) + (y^2 - 6y + 9) + (z^2 + 6z + 9) = -45 + 36 + 9 + 9\n  

  • Step 4: Write in standard form:\n  (x - 6)^2 + (y - 3)^2 + (z + 3)^2 = 9\n  

  • Center: $(6, 3, -3)$

  • Radius: $r = \sqrt{9} = 3$

• Intersections of a Sphere with Coordinate Planes:

  • Given sphere: $(x - 3)^2 + (y - 2)^2 + (z - 5)^2 = 9$

  • Intersection with $xz$-plane ($y = 0$):\n    (x - 3)^2 + (0 - 2)^2 + (z - 5)^2 = 9 \implies (x - 3)^2 + 4 + (z - 5)^2 = 9 \implies (x - 3)^2 + (z - 5)^2 = 5\n    

  • Geometric result: A circle in the $xz$-plane centered at $(3, 5)$ with radius $r = \sqrt{5}$. (Sphere slices through the plane).

  • Intersection with $yz$-plane ($x = 0$):\n    (0 - 3)^2 + (y - 2)^2 + (z - 5)^2 = 9 \implies 9 + (y - 2)^2 + (z - 5)^2 = 9 \implies (y - 2)^2 + (z - 5)^2 = 0\n    

  • Geometric result: A single point $(0, 2, 5)$. (Sphere is tangent to the $yz$-plane).

  • Intersection with $xy$-plane ($z = 0$):\n    (x - 3)^2 + (y - 2)^2 + (0 - 5)^2 = 9 \implies (x - 3)^2 + (y - 2)^2 + 25 = 9 \implies (x - 3)^2 + (y - 2)^2 = -16\n    

  • Geometric result: No real solution (radius squared cannot be negative). (Sphere hangs floating above the $xy$-plane without touching).

• Bounded Regions in $R^3$:

  • Region defined by $1 \le x^2 + y^2 + z^2 \le 9$ with condition $z \le 0$

  • $x^2 + y^2 + z^2 = 1$ is an inner solid sphere boundary of radius $1$ centered at origin.

  • $x^2 + y^2 + z^2 = 9$ is an outer solid sphere boundary of radius $3$ centered at origin.

  • $z \le 0$ restricts the domain to space lying on or below the $xy$-plane.

  • Geometric description: A solid, lower-hemispherical shell bounded between concentric spheres of radius $1$ and radius $3$.

Introduction to Vectors and Vector Operations

• Definitions & Properties:

  • Vector: A mathematical quantity possessing both magnitude (length) and direction (e.g., displacement, velocity, acceleration, force, force fields).

  • Scalar: A mathematical quantity possessing magnitude only (e.g., count of items in a bag).

• Notations:

  • Standard vector notation: $\mathbf{a}$, $\vec{a}$, or $\bar{a}$.

  • Component Form: $\mathbf{a} = \langle a_x, a_y, a_z \rangle$

  • Standard Basis Vector Form: $\mathbf{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}$, where $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ are unit vectors along $x$, $y$, $z$ axes.

• 2D Vector Breakdown:

  • $\mathbf{a} = \langle a_x, a_y \rangle = a_x \mathbf{i} + a_y \mathbf{j}$

  • $a_x = ||\mathbf{a}|| \cos(\theta)$

  • $a_y = ||\mathbf{a}|| \sin(\theta)$

• 3D Magnitude (Length) Formula:\n  ||\mathbf{a}|| = \sqrt{a_x^2 + a_y^2 + a_z^2}\n  

• Vector Directed Between Two Points:

  • For initial point $A(x_1, y_1, z_1)$ and terminal point $B(x_2, y_2, z_2)$, vector $\vec{AB}$ is defined as terminal coordinates minus initial coordinates:\n  \vec{AB} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j} + (z_2 - z_1)\mathbf{k}\n  

  • Reversing direction flips sign: $\vec{BA} = -\vec{AB}$.

• Equivalent/Equal Vectors:

  • Two vectors are equal if and only if they possess identical magnitudes AND identical directions, regardless of their position in space.

• Algebraic and Visual Vector Operations:

  • Vector Addition ($\mathbf{a} + \mathbf{b}$):

  • Algebraic: $\langle a_x + b_x, a_y + b_y, a_z + b_z \rangle$

  • Visual: Head-to-tail triangle law.

  • Scalar Multiplication ($c\mathbf{a}$):

  • Scales magnitude by factor $|c|$.

  • If $c > 1$: Stretches length.

  • If $0 < c < 1$: Shrinks length.

  • If $c < 0$: Reverses direction.

  • Vector Subtraction ($\mathbf{a} - \mathbf{b}$):

  • Algebraic: $\langle a_x - b_x, a_y - b_y, a_z - b_z \rangle$

  • Visual: Equivalent to adding the reversed vector $\mathbf{a} + (-1)\mathbf{b}$.

• Three Types of Multiplication Involving Vectors:

  1. Scalar Multiplication: Product of a scalar number and a vector (yields a scaled vector).

  2. Dot Product: Product of two vectors (yields a scalar quantity).

  3. Cross Product: Product of two vectors (yields a new vector orthogonal to both; exists exclusively in 3D space).

Questions, Dialogue & Student Interaction

• In-Class Instructor Q&A:

  • Student Question: What is the exam format going to be?

  • Instructor Answer: Finalized exam draft is pending. Exam will consist of multiple-choice and workout sections designed for a 75-minute block, coordinated across all course sections.

  • Student Question: Will practice exams be provided?

  • Instructor Answer: Yes, coordinated practice exams with answer keys will be uploaded as study resources.