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.

25

22 hrs

Payment required for enrolment
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.
[OAU, Ife] MTH 201: Mathematical Methods I
[OAU, Ife] MTH 201: Mathematical Methods I
This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

[NUC Core] GET 209: Engineering Mathematics I
[NUC Core] 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.

Course Chapters

1. Introduction
4

Meaning of real-valued functions of several variables - domain, range, graphs, level curves, surfaces and computer-aided visualizations.

Chapter lessons

1-1. Welcome
12:30

Welcome to the course and course outline.

1-2. Multivariable functions
15:35

Meaning and examples of multivariable real-valued functions, in contrast to single-variable ones.

1-3. Visualization
8:52

Graphing two-variable functions, in contrast to single-variable ones.

1-4. Single-valued functions
11:56

Meaning and examples of single-valued multivariable functions.

2. Limits
3
5

Formal and informal definitions, evaluation of limits of functions of several variables.

Chapter lessons

2-1. Definition
17:14

Formal and informal definitions of limits of functions of two variables.

2-2. Existence
16:50

Conditions for the existence of the limit of a function of several variables.

2-3. Procedure
20:02

General procedure for evaluating limits of functions of several variables.

3. Continuity
1
2

Continuity of functions of several variables.

Chapter lessons

3-1. Definition
9:58

Formal and informal definitions of continuity of two-variable real-valued functions.

4. Partial Derivatives
3
4

Formal definition of differentiability for functions of several variables; notations and evaluation of first and higher-order partial derivatives.

Chapter lessons

4-1. First partial derivatives
17:54

Meaning of the first partial derivatives of a function of two variables.

4-2. Higher-order partial derivatives
10:56

Meaning of higher-order partial derivatives of multivariable functions.

4-3. Notations
11:01

Notations for first and higher-order partial derivatives.

5. Composite Functions
1
3

Partial derivatives of composite functions of several variables - chain rule of differentiation.

Chapter lessons

5-1. Chain rule
27:31

Chain rule of differentiation for partial derivatives.

6. Implicit Differentiation
4
4

Differentiation of implicitly-defined functions of several variables and an introduction to the Jacobian determinant.

Chapter lessons

6-1. Two variables
28:29

Implicit differentiation of a function with one dependent variable and one independent variable using partial derivatives.

6-2. Jacobian Determinants
6:38

Definition and evaluation of Jacobian determinants.

6-3. Three variables
24:46

Implicit differentiation of a function with two dependent variables and one independent variable using partial derivatives.

6-4. Several variables
11:13

General implicit differentiation of a function with several dependent and independent variables using Jacobian determinants.

7. Theorems on Jacobians
3
3

Some theorems on Jacobian determinants - the implicit function theorem and its implications.

Chapter lessons

7-1. Existence of solutions
19:50

Examining the existence of solutions of a linear system of equations and its relation to the Jacobian determinant.

7-2. The implicit function theorem
11:50

The implicit function for a system of implicit functions.

7-3. Other theorems
11:53

Other theorems on the Jacobian determinant.

8. Homogeneous Functions
2
2

Meaning of homogeneous functions; Euler's theorem for homogeneous functions.

Chapter lessons

8-1. Definition
10:01

Meaning of homogeneous functions - identifying the order of homogeneous functions.

8-2. Euler's theorem
7:47

Euler's theorem for homogeneous functions.

9. Taylor's Theorem
1
2

Taylor's theorem for functions of two variables.

Chapter lessons

9-1. Theorem
22:44

Taylor's theorem for a function of two variables.

10. Extreme Values
2
2

Getting the stationary points of a function of several variables by application of partial derivatives.

Chapter lessons

10-1. Maxima and minima
28:34

How to find the stationary points of a function of two variables.

10-2. Procedure
7:26

How to determine the stationary points of a two-variable function - and the nature of the stationary points.

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.

Chapter lessons

11-1. Method
6:57

The method of Lagrange multiplier for examining stationary points of a function of two variables subject to a constraint.

12. Gradients and Directional Derivatives
2
2

Applications of partial derivatives - gradient of a differentiable function and its relation to the directional derivative.

Chapter lessons

12-1. The gradient vector
8:03

Definition of the gradient vector for a function of several variables.

12-2. Directional derivative
22:15

Definition of the directional derivative of a function of several variables, its maximum value and what direction it occurs.

13. Applications to Geometry
7
3

Equations of lines, planes, curves and surfaces in three-dimensional Cartesian coordinates - vector, standard and parametric equations.

Chapter lessons

13-1. Equation of a straight line
39:41

Vector (three-dimensional) equations of a straight line, and why y=mx+c no longer cuts it.

13-2. Equation of a plane
20:43

Equation of a plane - vector and standard forms.

13-3. Equation of a curve
15:10

Derivation and visualization of the parametric equations of curves in three dimensions and how they are related to those of straight lines; direction of the derivative of the parametric equation of a curve.

13-4. Equation of a surface
17:19

Equation of a surface, in contrast to a plane; the direction of the gradient vector of a surface at a given point.

13-5. Intersection of surfaces
26:52

Lines formed by intersection of surfaces and their tangent vector.

13-6. Tangent line and normal plane to a curve
29:42

Equations of the tangent line and normal plane to a curve.

13-7. Tangent plane and normal line to a surface
11:46

Equations of the tangent plane and normal line to a surface.