Mastering Calculus Iii - From Vectors To Theorems

Learn the core concepts of Calculus III with intuitive visuals, examples, and practical problem-solving.

What you'll learn

Master Vector Operations: Understand and apply dot product, cross product, projections, torque, and vector equations of lines and planes in 3D space.

isualize Multivariable Functions: Analyze domains, level curves, space curves, and tangent lines of multivariable functions.

Compute Partial Derivatives: Learn how to differentiate multivariable functions, including second-order and directional derivatives.

Apply Gradient Concepts: Use gradients to determine directions of steepest ascent/descent and solve optimization problems.

Derive Tangent Planes & Approximations: Construct tangent planes and use linear approximations for multivariable functions.

Identify Critical Points & Optimize: Find local extrema using second derivative tests and solve constrained optimization problems with Lagrange multipliers.

Evaluate Double & Triple Integrals: Set up and solve integrals in rectangular, polar, cylindrical, and spherical coordinates.

Transform Coordinates with Jacobians: Understand coordinate transformations and apply Jacobians in multivariable integration.

Analyze Vector Fields & Line Integrals: Compute circulation, flux, and potential functions of vector fields.

Apply Advanced Theorems: Use Green’s, Stokes’, and Divergence Theorems to evaluate line and surface integrals in vector calculus.

Requirements

Solid understanding of Calculus I & II (limits, derivatives, integrals)

Basic algebra and trigonometry skills

Description

Are you ready to conquer Calculus 3 with confidence? Whether you're a university student, STEM major, or lifelong learner, this comprehensive course will guide you through the essential topics of multivariable calculus—from vectors and partial derivatives to triple integrals and advanced theorems.Calculus 3—also known as multivariable calculus—is one of the most powerful tools in mathematics, engineering, physics, and data science. But for many students, it can feel overwhelming. This course is designed to demystify Calculus 3 by breaking down complex concepts into clear, visual, and practical lessons.Whether you're a university student preparing for exams, a STEM major needing a solid foundation, or a curious learner looking to understand the math behind 3D modeling, optimization, and physical systems—this course is for you.This course is designed to make complex concepts intuitive and accessible. With 60+ bite-sized lectures, visual explanations, and real-world applications, you'll build a strong foundation in higher-level calculus and be fully prepared for exams, engineering problems, or advanced studies.What Makes This Course Different?Structured Learning Path: Organized into 10 logical sections, each building on the last—from vectors to advanced theorems.Visual & Intuitive Explanations: Concepts are taught with diagrams, animations, and real-world analogies to make learning stick.Worked Examples: Every major topic includes step-by-step problem solving so you can follow along and practice.Preview-Enabled Lectures: Try before you commit—several introductory lessons are open for preview.Exam-Ready Preparation: Ideal for students taking university-level Calculus III or preparing for standardized tests.What You’ll LearnHow to work with vectors in 3D space, including dot and cross productsHow to analyze multivariable functions and space curvesHow to compute partial derivatives, gradients, and directional derivativesHow to find tangent planes and use linear approximationsHow to identify critical points and solve optimization problemsHow to evaluate double and triple integrals in various coordinate systemsHow to use Jacobians for coordinate transformationsHow to analyze vector fields and compute line integralsHow to apply Green’s, Stokes’, and Divergence TheoremsHow to connect theory to real-world applications in physics and engineering

Overview

Section 1: Vectors & Vector Operations

Lecture 1 An Introduction to Vectors

Lecture 2 How to Find the Unit Vector of a Vector

Lecture 3 Perpendicular and Parallel Vectors

Lecture 4 Solving for Angles and Components of Vectors

Lecture 5 Solving for Dot Product

Lecture 6 How to Find the Type of Angle Between Two Vectors

Lecture 7 Understanding Projections and Scalars on Vectors

Lecture 8 Solving for and Understanding Cross Product

Lecture 9 Using Cross Product to Find the Area of a Parallelogram/Triangle

Lecture 10 Solving for Torque Using Cross Product

Lecture 11 Finding the Angle Between Three Component Vectors Using Cross Product

Lecture 12 Solving for Angle Between Three Component Vectors Using Dot Product

Lecture 13 Understanding Vector Equation of Lines

Lecture 14 Finding The Intersecting Point of Two Vector Equation Lines

Lecture 15 Finding the Smallest Distance Between a Point and a Line Using Vectors

Lecture 16 How to Write the Equation of a Plane in 3D

Lecture 17 Easy Way to Find the Angle Between Two Planes

Lecture 18 Solving for the Equation of a Plane Given Three Points

Lecture 19 Equation of a Plane Given Two Vectors and a Point

Lecture 20 Equation of Line Intersecting Two Planes

Section 2: Multivariable Functions & Graphs

Lecture 21 Intro To Multivariable Functions/Graphs

Lecture 22 XY XZ and YZ Plane Intersects of a Multivariable Function

Lecture 23 What are Space Curves/Vector Valued Functions?

Lecture 24 Finding the Arc Length of Space Curves

Lecture 25 Equation of Tangent Line on a Space Curve

Lecture 26 Finding the Domain of Multivariable Functions

Lecture 27 Level Curves with Desmos - Calculus 3

Lecture 28 Multivariable Limits with Worked Examples - Calculus 3

Section 3: Partial Derivatives & Gradient

Lecture 29 How to Partially Differentiate with Worked Examples - Calculus 3

Lecture 30 Introduction to Second Order Partial Derivatives - Calculus 3

Lecture 31 How to Solve Directional Derivatives with Worked Examples - Calculus 3

Lecture 32 Introduction to the Gradient - Calculus 3

Lecture 33 Direction and Slope of Steepest Ascent/Descent with the Gradient - Calculus 3

Section 4: Tangent Planes & Approximations

Lecture 34 The Equation of a Tangent Plane with Worked Examples - Calculus 3

Lecture 35 Linear Approximation with Tangent Planes - Calculus 3

Section 5: Critical Points & Optimization

Lecture 36 How to Find Critical Points/Local Extrema - Calc III

Lecture 37 How to Use the Second Derivative Test - Calculus 3

Lecture 38 How to Find the Absolute Max/Min Using Parameterization of Boundary - Calculus 3

Lecture 39 Methods of Lagrange Multipliers Step-By-Step Explanations

Lecture 40 How to Find the Shortest Distance From A Point and A Plane Using Lagrange Multip

Section 6: Double Integrals

Lecture 41 How to Solve Double Integrals - Calculus 3

Lecture 42 Difference Between Calc I Integrals and Calc III Double Integrals

Lecture 43 How to Find the Average of a Double Integral

Lecture 44 How to Solve Double Integrals with Functions as Bounds

Lecture 45 How to Switch Bounds For Double Integrals

Lecture 46 How to Solve Double Integrals with Polar Coordinates

Section 7: Triple Integrals & Coordinate Systems

Lecture 47 Basic Introduction to Setting Up Triple Integrals

Lecture 48 Basic Introduction to Cylindrical Coordinates

Lecture 49 Basic Introduction to Spherical Coordinates

Lecture 50 How to Solve Triple Integrals

Lecture 51 How to Solve Triple Integrals in Cylindrical Coordinates

Lecture 52 How to Solve Triple Integrals in Spherical Coordinates

Section 8: Jacobian & Coordinate Transformations

Lecture 53 Why Does dA = rdrdθ - Intro to Jacobians

Lecture 54 How to Find A Jacobian for Triple Integrals - Example Spherical Coordinates

Lecture 55 How to Solve Integrals Using Coordinate Transformations

Section 9: Vector Fields & Line Integrals

Lecture 56 Introduction to Vector Fields

Lecture 57 How to Solve Line Integrals of Scalar Functions

Lecture 58 How to Find the Circulation of a Vector Line Integral

Lecture 59 How to Find the Flux of a Vector Line Integral

Lecture 60 The Fundamental Theorem of Calculus - Calc III

Lecture 61 How to Find the Potential Function of a Conservative Vector Field

Section 10: Curl, Divergence & Theorems

Lecture 62 Finding and Getting an Intuition for 2D and 3D Curl

Lecture 63 Everything You Need to Know About Divergence in 7 Minutes

Lecture 64 How to Solve Line Integrals Using Green's Theorem

Lecture 65 How to Solve Surface Integrals of Scalar Functions

Lecture 66 How to Solve Vector Surface Integrals

Lecture 67 How to Use Stokes' Theorem

Lecture 68 How to Use Divergence Theorem

Lecture 69 All Calc III Theorems and Vector Integrals In a Nutshell

University students taking Calculus III or Multivariable Calculus,Engineering, physics, and math majors,Anyone preparing for standardized tests or advanced STEM courses,Curious learners who want to understand the math behind 3D modeling, physics, and optimization