**Course overview**

Learn multivariable calculus with this **Calculus 3 (Multivariable) Masterclass** course. In this course, you will understand the complexities of multivariable calculus and solve various problems.

The **Calculus 3 (Multivariable) Masterclass** course will go over the fundamental ideas about multivariable functions. It will introduce you to analytical geometry and explain where it’s used. You will identify the distance formula to calculate distance between two points and learn how to calculate dot and cross products. In addition, you will learn about conic sections, topology, partial derivatives and vector-valued functions and use them to solve problems. Many practice problems with solutions are included in this course to improve your problem-solving abilities.

**Learning outcomes**

- Learn about limit, continuity and differentiability
- Know how to graph a parabola using conic sections
- Understand what is a paraboloid in calculus
- Identify the differentiation rules and use them to solve problems
- Learn about different coordinate systems
- Be able to compute a composite function’s derivative using chain rule
- Learn how to use Taylor’s formula

**Who Is This Course For?**

Anyone interested in learning Calculus most efficiently can take this **Calculus 3 (Multivariable) Masterclass** course. The skills gained from this training will provide excellent opportunities for career advancement.

**Entry Requirement**

- This course is available to all learners of all academic backgrounds.
- Learners should be aged 16 or over.
- Good understanding of English language, numeracy and ICT skills are required to take this course.

**Certification**

- After you have successfully completed the course, you will obtain an Accredited Certificate of Achievement. And, you will also receive a Course Completion Certificate following the course completion without sitting for the test. Certificates can be obtained either in hardcopy for £39 or in PDF format at the cost of £24.
- PDF certificate’s turnaround time is 24 hours, and for the hardcopy certificate, it is 3-9 working days.

**Why Choose Us?**

- Affordable, engaging & high-quality e-learning study materials;
- Tutorial videos and materials from the industry-leading experts;
- Study in a user-friendly, advanced online learning platform;
- Efficient exam systems for the assessment and instant result;
- United Kingdom & internationally recognized accredited qualification;
- Access to course content on mobile, tablet and desktop from anywhere, anytime;
- Substantial career advancement opportunities;
- 24/7 student support via email.

**Career Path**

The **Calculus 3 (Multivariable) Masterclass** course provides essential skills that will make you more effective in your role. It would be beneficial for any related profession in the industry, such as:

- Math’s Teacher

### Course Curriculum

Unit 1 - About the course | |||

Introduction to the course | 00:30:00 | ||

Unit 2 - Analytical geometry in the space | |||

The plane R^2 and the 3-space R^3: points and vectors | 00:25:00 | ||

Distance between points | 00:08:00 | ||

Vectors and their products | 00:04:00 | ||

Dot product | 00:14:00 | ||

Cross product | 00:13:00 | ||

Scalar triple product | 00:07:00 | ||

Describing reality with numbers; geometry and physics | 00:06:00 | ||

Straight lines in the plane | 00:08:00 | ||

Planes in the space | 00:13:00 | ||

Straight lines in the space | 00:08:00 | ||

Unit 3 - Conic Units: circle, ellipse, parabola, hyperbola | |||

Conic Units, an introduction | 00:06:00 | ||

Quadratic curves as conic Units | 00:10:00 | ||

Definitions by distance | 00:17:00 | ||

Cheat sheets | 00:04:00 | ||

Circle and ellipse, theory | 00:19:00 | ||

Parabola and hyperbola, theory | 00:12:00 | ||

Completing the square | 00:04:00 | ||

Completing the square, problems 1 and 2 | 00:12:00 | ||

Completing the square, problem 3 | 00:10:00 | ||

Completing the square, problems 4 and 5 | 00:08:00 | ||

Completing the square, problems 6 and 7 | 00:08:00 | ||

Unit 4 - Quadric surfaces: spheres, cylinders, cones, ellipsoids, paraboloids etc | |||

Quadric surfaces, an introduction | 00:16:00 | ||

Degenerate quadrics | 00:17:00 | ||

Ellipsoids | 00:08:00 | ||

Paraboloids | 00:16:00 | ||

Hyperboloids | 00:25:00 | ||

Problems 1 and 2 | 00:09:00 | ||

Problem 3 | 00:07:00 | ||

Problems 4 and 5 | 00:10:00 | ||

Problem 6 | 00:06:00 | ||

Unit 5 - Topology in R^n | |||

Neighborhoods | 00:07:00 | ||

Open, closed, and bounded sets | 00:14:00 | ||

Identify sets, an introduction | 00:04:00 | ||

Example 1 | 00:06:00 | ||

Example 2 | 00:06:00 | ||

Example 3 | 00:05:00 | ||

Example 4 | 00:06:00 | ||

Example 5 | 00:04:00 | ||

Example 6 and 7 | 00:06:00 | ||

Unit 6 - Coordinate systems | |||

Different coordinate systems | 00:02:00 | ||

Polar coordinates in the plane | 00:11:00 | ||

An important example | 00:07:00 | ||

Solving 3 problems | 00:19:00 | ||

Cylindrical coordinates in the space | 00:03:00 | ||

Problem 1 | 00:03:00 | ||

Problem 2 | 00:02:00 | ||

Problem 3 | 00:04:00 | ||

Problem 4 | 00:04:00 | ||

Spherical coordinates in the space | 00:08:00 | ||

Some examples | 00:08:00 | ||

Conversion | 00:08:00 | ||

Problem 1 | 00:08:00 | ||

Problem 2 | 00:12:00 | ||

Problem 3 | 00:11:00 | ||

Problem 4 | 00:07:00 | ||

Unit 7 - Vector-valued functions, introduction | |||

Curves: an introduction | 00:10:00 | ||

Functions: repetition | 00:08:00 | ||

Functions: repetition | 00:08:00 | ||

Vector-valued functions, parametric curves: domain | 00:08:00 | ||

Unit 8 - Some examples of parametrisation | |||

Vector-valued functions, parametric curves | 00:11:00 | ||

An intriguing example | 00:14:00 | ||

Problem 1 | 00:12:00 | ||

Problem 2 | 00:13:00 | ||

Problem 3 | 00:15:00 | ||

Problem 4, helix | 00:09:00 | ||

Unit 9 - Vector-valued calculus; curve: continuous, differentiable, and smooth | |||

Notation | 00:05:00 | ||

Limit and continuity | 00:09:00 | ||

Derivatives | 00:14:00 | ||

Speed, acceleration | 00:08:00 | ||

Position, velocity, acceleration: an example | 00:06:00 | ||

Smooth and piecewise smooth curves | 00:09:00 | ||

Sketching a curve | 00:15:00 | ||

Sketching a curve: an exercise | 00:16:00 | ||

Example 1 | 00:11:00 | ||

Example 2 | 00:16:00 | ||

Example 3 | 00:10:00 | ||

Extra theory: limit and continuity | 00:19:00 | ||

Extra theory: derivative, tangent, and velocity | 00:13:00 | ||

Differentiation rules | 00:27:00 | ||

Differentiation rules, example 1 | 00:19:00 | ||

Differentiation rules: example 2 | 00:19:00 | ||

Position, velocity, acceleration, example 3 | 00:15:00 | ||

Position and velocity, one more example | 00:15:00 | ||

Trajectories of planets | 00:13:00 | ||

Unit 10 - Arc length | |||

Parametric curves: arc length | 00:15:00 | ||

Arc length: problem 1 | 00:11:00 | ||

Arc length: problems 2 and 3 | 00:15:00 | ||

Arc length: problems 4 and 5 | 00:13:00 | ||

Unit 11 - Arc length parametrisation | |||

Parametric curves: parametrisation by arc length | 00:10:00 | ||

Parametrisation by arc length, how to do it, example 1 | 00:12:00 | ||

Parametrisation by arc length, example 2 | 00:22:00 | ||

Arc length does not depend on parametrisation, theory | 00:14:00 | ||

Unit 12 - Real-valued functions of multiple variables | |||

Functions of several variables, introduction | 00:09:00 | ||

Introduction, continuation 1 | 00:14:00 | ||

Introduction, continuation 2 | 00:08:00 | ||

Domain | 00:06:00 | ||

Domain, problem solving part 1 | 00:18:00 | ||

Domain, problem solving part 2 | 00:13:00 | ||

Domain, problem solving part 3 | 00:15:00 | ||

Functions of several variables, graphs | 00:14:00 | ||

Plotting functions of two variables, problems part 1 | 00:16:00 | ||

Plotting functions of two variables, problems part 2 | 00:12:00 | ||

Level curves | 00:14:00 | ||

Level curves, problem 1 | 00:10:00 | ||

Level curves, problem 2 | 00:08:00 | ||

Level curves, problem 3 | 00:09:00 | ||

Level curves, problem 4 | 00:14:00 | ||

Level curves, problem 5 | 00:16:00 | ||

Level surfaces, definition and problem solving | 00:20:00 | ||

Unit 13 - Limit, continuity | |||

Limit and continuity, part 1 | 00:18:00 | ||

Limit and continuity, part 2 | 00:15:00 | ||

Limit and continuity, part 3 | 00:20:00 | ||

Problem solving 1 | 00:25:00 | ||

Problem solving 2 | 00:18:00 | ||

Problem solving 3 | 00:20:00 | ||

Problem solving 4 | 00:15:00 | ||

Unit 14 - Partial derivative, tangent plane, normal line, gradient, Jacobian | |||

Introduction 1: definition and notation | 00:10:00 | ||

Introduction 2: arithmetical consequences | 00:12:00 | ||

Introduction 3: geometrical consequences (tangent plane) | 00:13:00 | ||

Introduction 4: partial derivatives not good enough | 00:06:00 | ||

Introduction 5: a pretty terrible example | 00:15:00 | ||

Tangent plane, part 1 | 00:07:00 | ||

Normal vector | 00:15:00 | ||

Tangent plane part 2: normal equation | 00:09:00 | ||

Normal line | 00:08:00 | ||

Tangent planes, problem 1 | 00:14:00 | ||

Tangent planes, problem 2 | 00:13:00 | ||

Tangent planes, problem 3 | 00:16:00 | ||

Tangent planes, problem 4 | 00:09:00 | ||

Tangent planes, problem 5 | 00:11:00 | ||

The gradient | 00:11:00 | ||

A way of thinking about functions from R^n to R^m | 00:11:00 | ||

The Jacobian | 00:14:00 | ||

Unit 15 - Higher partial derivatives | |||

Introduction | 00:15:00 | ||

Definition and notation | 00:07:00 | ||

Mixed partials, Hessian matrix | 00:13:00 | ||

The difference between Jacobian matrices and Hessian matrices | 00:08:00 | ||

Equality of mixed partials; Schwarz’ theorem | 00:09:00 | ||

Schwarz’ theorem: Peano’s example | 00:06:00 | ||

Schwarz’ theorem: the proof | 00:19:00 | ||

Partial Differential Equations, introduction | 00:04:00 | ||

Partial Differential Equations, basic ideas | 00:11:00 | ||

Partial Differential Equations, problem solving | 00:13:00 | ||

Laplace equation and harmonic functions 1 | 00:08:00 | ||

Laplace equation and harmonic functions 2 | 00:06:00 | ||

Laplace equation and Cauchy-Riemann equations | 00:11:00 | ||

Dirichlet problem | 00:07:00 | ||

Unit 16 - Chain rule: different variants | |||

A general introduction | 00:17:00 | ||

Variants 1 and 2 | 00:10:00 | ||

Variant 3 | 00:18:00 | ||

Variant 3 (proof) | 00:11:00 | ||

Variant 4 | 00:09:00 | ||

Example with a diagram | 00:04:00 | ||

Problem solving | 00:08:00 | ||

Problem solving, problem 1 | 00:04:00 | ||

Problem solving, problem 2 | 00:09:00 | ||

Problem solving, problem 3 | 00:33:00 | ||

Problem solving, problem 4 | 00:15:00 | ||

Problem solving, problem 5 | 00:28:00 | ||

Problem solving, problem 6 | 00:09:00 | ||

Problem solving, problem 7 | 00:06:00 | ||

Problem solving, problem 8 | 00:18:00 | ||

Unit 17 - Linear approximation, linearisation, differentiability, differential | |||

Linearisation and differentiability in Calc1 | 00:11:00 | ||

Differentiability in Calc3: introduction | 00:15:00 | ||

Differentiability in two variables, an example | 00:10:00 | ||

Differentiability in Calc3 implies continuity | 00:10:00 | ||

Partial differentiability does NOT imply differentiability | 00:05:00 | ||

An example: continuous, not differentiable | 00:06:00 | ||

Differentiability in several variables, a test | 00:18:00 | ||

Differentiability, Partial Differentiability, and Continuity in Calc3 | 00:12:00 | ||

Differentiability in two variables, a geometric interpretation | 00:11:00 | ||

Linearization: two examples | 00:16:00 | ||

Linearization, problem solving 1 | 00:11:00 | ||

Linearization, problem solving 2 | 00:11:00 | ||

Linearization, problem solving 3 | 00:12:00 | ||

Linearization by Jacobian matrix, problem solving | 00:16:00 | ||

Differentials: problem solving 1 | 00:11:00 | ||

Differentials: problem solving 2 | 00:10:00 | ||

Unit 18 - Gradient, directional derivatives | |||

Gradient | 00:04:00 | ||

The gradient in each point is orthogonal to the level curve through the point | 00:08:00 | ||

The gradient in each point is orthogonal to the level surface through the point | 00:14:00 | ||

Tangent plane to the level surface, an example | 00:06:00 | ||

Directional derivatives, introduction | 00:06:00 | ||

Directional derivatives, the direction | 00:04:00 | ||

How to normalize a vector and why it works | 00:08:00 | ||

Directional derivatives, the definition | 00:07:00 | ||

Partial derivatives as a special case of directional derivatives | 00:05:00 | ||

Directional derivatives, an example | 00:11:00 | ||

Directional derivatives: important theorem for computations and interpretations | 00:10:00 | ||

Directional derivatives: an earlier example revisited | 00:05:00 | ||

Geometrical consequences of the theorem about directional derivatives | 00:10:00 | ||

Geometical consequences of the theorem about directional derivatives, an example | 00:07:00 | ||

Directional derivatives, an example | 00:11:00 | ||

Normal line and tangent line to a level curve: how to get their equations | 00:06:00 | ||

Normal line and tangent line to a level curve: their equations, an example | 00:14:00 | ||

Gradient and directional derivatives, problem 1 | 00:18:00 | ||

Gradient and directional derivatives, problem 2 | 00:20:00 | ||

Gradient and directional derivatives, problem 3 | 00:09:00 | ||

Gradient and directional derivatives, problem 4 | 00:04:00 | ||

Gradient and directional derivatives, problem 5 | 00:12:00 | ||

Gradient and directional derivatives, problem 6 | 00:10:00 | ||

Gradient and directional derivatives, problem 7 | 00:13:00 | ||

Unit 19 - Implicit functions | |||

What is the Implicit Function Theorem? | 00:13:00 | ||

Jacobian determinant | 00:04:00 | ||

Jacobian determinant for change to polar and to cylindrical coordinates | 00:07:00 | ||

Jacobian determinant for change to spherical coordinates | 00:09:00 | ||

Jacobian determinant and change of area | 00:10:00 | ||

The Implicit Function Theorem variant 1 | 00:08:00 | ||

The Implicit Function Theorem variant 1, an example | 00:15:00 | ||

The Implicit Function Theorem variant 2 | 00:10:00 | ||

The Implicit Function Theorem variant 2, example 1 | 00:07:00 | ||

The Implicit Function Theorem variant 2, example 2 | 00:14:00 | ||

The Implicit Function Theorem variant 3 | 00:15:00 | ||

The Implicit Function Theorem variant 3, an example | 00:12:00 | ||

The Implicit Function Theorem variant 4 | 00:11:00 | ||

The Inverse Function Theorem | 00:09:00 | ||

The Implicit Function Theorem, summary | 00:04:00 | ||

Notation in some unclear cases | 00:08:00 | ||

The Implicit Function Theorem, problem solving 1 | 00:27:00 | ||

The Implicit Function Theorem, problem solving 2 | 00:13:00 | ||

The Implicit Function Theorem, problem solving 3 | 00:07:00 | ||

The Implicit Function Theorem, problem solving 4 | 00:16:00 | ||

Unit 20 - Taylor’s formula, Taylor’s polynomial, quadratic forms | |||

Taylor’s formula, introduction | 00:10:00 | ||

Quadratic forms and Taylor’s polynomial of second degree | 00:22:00 | ||

Taylor’s polynomial of second degree, theory | 00:11:00 | ||

Taylor’s polynomial of second degree, example 1 | 00:07:00 | ||

Taylor’s polynomial of second degree, example 2 | 00:04:00 | ||

Taylor’s polynomial of second degree, example 3 | 00:11:00 | ||

Classification of quadratic forms (positive definite etc) | 00:12:00 | ||

Classification of quadratic forms, problem solving 1 | 00:08:00 | ||

Classification of quadratic forms, problem solving 2 | 00:14:00 | ||

Classification of quadratic forms, problem solving 3 | 00:10:00 | ||

Unit 21 - Optimization on open domains (critical points) | |||

Extreme values of functions of several variables | 00:12:00 | ||

Extreme values of functions of two variables, without computations | 00:10:00 | ||

Critical points and their classification (max, min, saddle) | 00:09:00 | ||

Second derivative test for C^3 functions of several variables | 00:12:00 | ||

Second derivative test for C^3 functions of two variables | 00:06:00 | ||

Critical points and their classification: some simple examples | 00:06:00 | ||

Critical points and their classification: more examples 1 | 00:05:00 | ||

Critical points and their classification: more examples 2 | 00:08:00 | ||

Critical points and their classification: more examples 3 | 00:10:00 | ||

Critical points and their classification: a more difficult example (4) | 00:47:00 | ||

Unit 22 - Optimization on compact domains | |||

Extreme values for continuous functions on compact domains | 00:06:00 | ||

Eliminate a variable on the boundary | 00:10:00 | ||

Parameterize the boundary | 00:08:00 | ||

Unit 23 - Lagrange multipliers (optimization with constraints) | |||

Lagrange multipliers 1 | 00:13:00 | ||

Lagrange multipliers 1, an old example revisited | 00:08:00 | ||

Lagrange multipliers 1, another example | 00:13:00 | ||

Lagrange multipliers 2 | 00:10:00 | ||

Lagrange multipliers 2, an example | 00:18:00 | ||

Lagrange multipliers 3 | 00:08:00 | ||

Lagrange multipliers 3, an example | 00:09:00 | ||

Summary: optimization | 00:07:00 | ||

Unit 24 - Final words | |||

The last one | 00:05:00 |

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