Partial Differentiation and Its Applications - Multivariable Calculus (Undergraduate Advanced)

This course provides a complete guide to the calculus of several variables. It builds from the foundational concepts of multivariable functions, limits, and continuity to the core techniques of differentiation, including partial derivatives, the chain rule, and implicit differentiation. The material culminates in advanced topics such as Taylor's theorem for several variables and the use of Jacobians. Multivariable calculus is the language of modern science, engineering, and economics. Its principles are used to model complex surfaces, analyse thermodynamic systems, create 3D computer graphics, and solve critical optimisation problems in business and finance. This is the mathematical toolkit for working with systems that have multiple interacting variables. By the end of this course, you will be able to calculate partial derivatives, apply the multivariable chain rule, and find directional derivatives using the gradient vector. You will also be able to solve both unconstrained and constrained optimisation problems by finding extreme values and using the method of Lagrange multipliers, and apply these derivatives to find tangent planes to surfaces. This course is for students who have completed a full single-variable calculus sequence. It is the standard curriculum for a multivariable calculus (Calculus III) module and is a direct prerequisite for the study of vector calculus, differential equations, and advanced courses in physics, engineering, and economics.

22 hrs

Enrolment valid for 12 months
This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.
GET 209: Engineering Mathematics I
GET 209: Engineering Mathematics I
Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

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MTH 201: Mathematical Methods I
MTH 201: Mathematical Methods I
Mastering advanced calculus is essential for modelling complex systems in science and engineering. This track delivers the rigorous mathematical foundation demanded by the official NUC CCMAS curriculum for MTH 201. It systematically builds your expertise from fundamental single-variable theory to the sophisticated multivariable analysis used to solve critical problems in physics, economics, and technology. This programme is for undergraduates in engineering, mathematics, physics, and computer science requiring a deep theoretical and practical command of calculus. It also serves economics students needing advanced quantitative tools or professionals in finance and data science seeking a solid mathematical base for technical research. You will gain the analytical skills to construct formal proofs for differentiation rules and apply cornerstone theorems like Mean Value and Taylor's. You will master multivariable techniques, enabling you to solve constrained optimization problems with Lagrange multipliers and compute multiple integrals across line, surface, and volume domains. This track is the requisite preparation for advanced studies in differential equations, vector analysis, and complex engineering modelling.

Mastering advanced calculus is essential for modelling complex systems in science and engineering. This track delivers the rigorous mathematical foundation demanded by the official NUC CCMAS curriculum for MTH 201. It systematically builds your expertise from fundamental single-variable theory to the sophisticated multivariable analysis used to solve critical problems in physics, economics, and technology. This programme is for undergraduates in engineering, mathematics, physics, and computer science requiring a deep theoretical and practical command of calculus. It also serves economics students needing advanced quantitative tools or professionals in finance and data science seeking a solid mathematical base for technical research. You will gain the analytical skills to construct formal proofs for differentiation rules and apply cornerstone theorems like Mean Value and Taylor's. You will master multivariable techniques, enabling you to solve constrained optimization problems with Lagrange multipliers and compute multiple integrals across line, surface, and volume domains. This track is the requisite preparation for advanced studies in differential equations, vector analysis, and complex engineering modelling.

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Course Chapters

1. Introduction
4
Meaning of real-valued functions of several variables - domain, range, graphs, level curves, surfaces and computer-aided visualizations.
Concept Overviews
4 Lessons
48:53
2. Limits
3
5
Formal and informal definitions, evaluation of limits of functions of several variables.
Concept Overviews
3 Lessons
54:06
Problem Walkthroughs
5 Lessons
1:24:38
3. Continuity
1
2
Continuity of functions of several variables.
Concept Overviews
1 Lesson
9:58
Problem Walkthroughs
2 Lessons
29:45
4. Partial Derivatives
3
4
Formal definition of differentiability for functions of several variables; notations and evaluation of first and higher-order partial derivatives.
Concept Overviews
3 Lessons
39:51
Problem Walkthroughs
4 Lessons
52:17
5. Composite Functions
1
3
Partial derivatives of composite functions of several variables - chain rule of differentiation.
Concept Overviews
1 Lesson
27:31
Problem Walkthroughs
3 Lessons
25:47
6. Implicit Differentiation
4
4
Differentiation of implicitly-defined functions of several variables and an introduction to the Jacobian determinant.
Concept Overviews
4 Lessons
1:11:06
Problem Walkthroughs
4 Lessons
47:15
7. Theorems on Jacobians
3
3
Some theorems on Jacobian determinants - the implicit function theorem and its implications.
Concept Overviews
3 Lessons
43:33
Problem Walkthroughs
3 Lessons
58:15
8. Homogeneous Functions
2
2
Meaning of homogeneous functions; Euler's theorem for homogeneous functions.
Concept Overviews
2 Lessons
17:48
Problem Walkthroughs
2 Lessons
32:50
9. Taylor's Theorem
1
2
Taylor's theorem for functions of two variables.
Concept Overviews
1 Lesson
22:44
Problem Walkthroughs
2 Lessons
35:26
10. Extreme Values
2
2
Getting the stationary points of a function of several variables by application of partial derivatives.
Concept Overviews
2 Lessons
36:00
Problem Walkthroughs
2 Lessons
43:40
11. Lagrange Multipliers
1
2
Getting the stationary points of a function of several variables subject to some condition by application of partial derivatives and the method of Lagrange multipliers.
Concept Overviews
1 Lesson
6:57
Problem Walkthroughs
2 Lessons
38:52
12. Gradients and Directional Derivatives
2
2
Applications of partial derivatives - gradient of a differentiable function and its relation to the directional derivative.
Concept Overviews
2 Lessons
30:18
Problem Walkthroughs
2 Lessons
25:37
13. Applications to Geometry
7
3
Equations of lines, planes, curves and surfaces in three-dimensional Cartesian coordinates - vector, standard and parametric equations.
Concept Overviews
7 Lessons
2:41:13
Problem Walkthroughs
3 Lessons
41:28