Partial Differentiation and Its Applications

Partial Differentiation and Its Applications - Advanced Calculus

This course introduces multi-variable real-valued functions and thoroughly addresses their limits, continuity, partial and total derivatives, applications and related concepts. You will learn how to: - Define and classify real-valued functions of several variables and their properties, such as domain, range, graphs, level curves and surfaces - Find the limit of a function of several variables as the independent variables approach certain values, and use the formal and informal definitions of limits - Find the continuity of a function of several variables at a point or on a set, and use the graphical, formal, and informal definitions of continuity - Find the partial derivative of a function of several variables with respect to one of the independent variables, and use the formal definition and various rules of partial differentiation - Find the higher-order partial derivatives of a function of several variables by applying the partial differentiation rules repeatedly, and use the notation and terminology for higher partial derivatives - Find the partial derivative of a function of several variables that is composed of other functions, and use the chain rule to differentiate composite functions - Find the partial derivative of a function of several variables that is homogeneous, and use the Euler's theorem to simplify the calculation - Find the total differential and derivative of a function of several variables, and use them to approximate the change in the function value - Find the partial derivative of a function of several variables that is defined implicitly by an equation - Find the Jacobian determinant of a function of several variables, and use it to obtain derivatives of implicit functions - Find the Taylor series expansion of a function of several variables, and use it to approximate the function value and its derivatives - Find the equation of the tangent plane and the normal line to a surface defined by a function of two variables, and use them to analyze the local behaviour of the surface - Find the extreme values of a function of several variables, and use the partial derivatives and the second derivative test to determine the nature of the extrema - Find the extreme values of a function of several variables subject to some constraint, and use the partial derivatives and the method of Lagrange multipliers to solve the constrained optimization problem This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid understanding of derivatives of functions of several variables. You will also be able to apply the knowledge and skills you gain to real-world problems and challenges that involve functions of several variables and their derivatives. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

22 hrs

$ 10.00

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

MTH 201: Mathematical Methods I

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

Introduction

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

Chapter lessons

1.Welcome12:30

Welcome to the course and course outline.

2.Multivariable functions15:35

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

3.Visualization8:52

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

4.Single-valued functions11:56

Meaning and examples of single-valued multivariable functions.

Limits

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

Chapter lessons

1.Definition17:14

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

2.Existence16:50

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

3.Procedure20:02

General procedure for evaluating limits of functions of several variables.

4.Worked examples (1)8:27

Worked examples on evaluating limits of two-variable real-valued functions.

5.Worked examples (2)16:56

More worked examples on evaluating limits of two-variable real-valued functions.

6.Worked examples (3)14:55

More worked examples on evaluating limits of two-variable real-valued functions.

7.Worked examples (4)15:36

More worked examples on evaluating limits of two-variable real-valued functions.

8.Worked examples (5)28:44

More worked examples on evaluating limits of two-variable real-valued functions.

Continuity

Continuity of functions of several variables.

Chapter lessons

1.Definition9:58

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

2.Worked examples (1)9:40

Worked examples on evaluating continuity of real-valued two-variable functions.

3.Worked examples (2)20:05

More worked examples on evaluating continuity of real-valued two-variable functions.

Partial Derivatives

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

Chapter lessons

1.First partial derivatives17:54

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

2.Higher-order partial derivatives10:56

Meaning of higher-order partial derivatives of multivariable functions.

3.Notations11:01

Notations for first and higher-order partial derivatives.

4.Worked examples (1)11:53

Worked examples on evaluating first partial derivatives from the first principles.

5.Worked examples (2)10:48

More worked examples on evaluating first partial derivatives from the first principles.

6.Worked examples (3)6:24

Worked examples on evaluating first and higher-order partial derivatives using the general methods of differentiation.

7.Worked examples (4)23:12

Worked examples on evaluating first and higher-order partial derivatives using the general methods of differentiation.

Composite Functions

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

Chapter lessons

1.Chain rule27:31

Chain rule of differentiation for partial derivatives.

2.Worked examples (1)6:11

Worked examples on partial derivatives of composite functions.

3.Worked examples (2)7:14

More worked examples on partial derivatives of composite functions.

4.Worked examples (3)12:22

More worked examples on partial derivatives of composite functions.

Implicit Differentiation

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

Chapter lessons

1.Two variables28:29

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

2.Jacobian Determinants6:38

Definition and evaluation of Jacobian determinants.

3.Three variables24:46

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

4.Several variables11:13

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

5.Worked examples (1)9:47

Worked examples on implicit differentiation of functions of several variables using partial derivatives and Jacobian determinants.

6.Worked examples (2)6:21

More worked examples on implicit differentiation of functions of several variables using partial derivatives and Jacobian determinants.

7.Worked examples (3)16:16

More worked examples on implicit differentiation of functions of several variables using partial derivatives and Jacobian determinants.

8.Worked examples (4)14:51

More worked examples on implicit differentiation of functions of several variables using partial derivatives and Jacobian determinants.

Theorems on Jacobians

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

Chapter lessons

1.Existence of solutions19:50

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

2.The implicit function theorem11:50

The implicit function for a system of implicit functions.

3.Other theorems11:53

Other theorems on the Jacobian determinant.

4.Worked examples (1)13:39

Worked examples on theorems on Jacobians.

5.Worked examples (2)22:40

More worked examples on theorems on Jacobians.

6.Worked examples (3)21:56

More worked examples on theorems on Jacobians.

Homogeneous Functions

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

Chapter lessons

1.Definition10:01

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

2.Euler's theorem7:47

Euler's theorem for homogeneous functions.

3.Worked examples (1)12:54

Worked examples on homogeneous functions and Euler's theorem.

4.Worked examples (2)19:56

More worked examples on homogeneous functions and Euler's theorem.

Taylor's Theorem

Taylor's theorem for functions of two variables.

Chapter lessons

1.Theorem22:44

Taylor's theorem for a function of two variables.

2.Worked examples (1)25:33

Worked examples on Taylor's theorem for a function of two variables.

3.Worked examples (2)9:53

More worked examples on Taylor's theorem for a function of two variables.

Extreme Values

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

Chapter lessons

1.Maxima and minima28:34

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

2.Procedure7:26

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

3.Worked examples (1)23:55

Worked examples on stationary points of a function of two variables.

4.Worked examples (2)19:45

More worked examples on stationary points of a function of two variables.

Lagrange Multipliers

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

1.Method6:57

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

2.Worked examples (1)21:04

Worked examples on the method of Lagrange multiplier for examining stationary points of a function of two variables subject to a constraint.

3.Worked examples (2)17:48

More worked examples on the method of Lagrange multiplier for examining stationary points of a function of two variables subject to a constraint.

Gradients and Directional Derivatives

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

Chapter lessons

1.The gradient vector8:03

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

2.Directional derivative22:15

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

3.Worked examples (1)11:31

Worked examples on gradients and directional derivatives.

4.Worked examples (2)14:06

More worked examples on gradients and directional derivatives.

Applications to Geometry

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

Chapter lessons

1.Equation of a straight line39:41

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

2.Equation of a plane20:43

Equation of a plane - vector and standard forms.

3.Equation of a curve15:10

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.

4.Equation of a surface17:19

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

5.Intersection of surfaces26:52

Lines formed by intersection of surfaces and their tangent vector.

6.Tangent line and normal plane to a curve29:42

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

7.Tangent plane and normal line to a surface11:46

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

8.Worked examples (1)12:08

Worked examples on equations of lines, planes, curves and surfaces in three dimensions.

9.Worked examples (2)10:32

Worked examples on equations of lines, planes, curves and surfaces in three dimensions.

10.Worked examples (3)18:48

More worked examples on equations of lines, planes, curves and surfaces in three dimensions.