## MATH20290: Multivariable Calculus for Engineers   --- Semester 1, 2013

### Lecturer: Masha Vlasenko

Office 3.45A, School of Mathematical Sciences
Belfield Office Park

The prerequisite to this course is single variable calculus. As its name suggests, multivariable calculus is the extension of calculus to more than one variable. That is, in single variable calculus one studies functions of a single independent variable y=f(x). In multivariable calculus we will study functions of two or more independent variables z=f(x, y), w=f(x, y, z), etc. These functions are essential for describing the physical world since many things depend on more than one independent variable. For example, in thermodynamics pressure depends on volume and temperature, in electricity and magnetism the magnetic and electric fields are functions of the three space variables (x,y,z) and one time variable t. Multivariable calculus is a highly geometric subject. We will relate graphs of functions to derivatives and integrals and see that visualization of graphs is harder but more rewarding and useful in several geometric dimensions.

The course begins with an introduction to analytic geometry in 2 and 3 dimensions. Then we study differential calculus of functions of several variables. The third part of the course deals with ordinary differential equations and Laplace transform.

FINAL EXAM took place on Saturday, December 14: [paper], [solutions], grading scheme
TO COLLECT YOUR HOMEWORK PAPERS: look at the MATH20290 box in the administration office of the School of Mathematical Sciences -- 5th floor of the library building (there are 3 boxes, where the tutors put homework).

Comments on this year's final exam paper can be found on the last page of the lecture notes. There will be NO FORMULAS given in the paper, just a table of a few Laplace transforms, sufficient to solve the respective question. GOOD LUCK AT THE EXAM!

MIDTERM EXAM took place on Monday, October 14: [paper] [solutions]
TUTORIALS take place in NT2 ART(in the basement of Newman Building), at 5pm on Tuesdays and Thursdays. In case you can't attend the tutorial you are assigned to, please attend the other one in the same week or submit your homework through someone else who attends the tutorial. If you have tutorials once in two weeks, please hand in two homeworks together, e.g. homeworks 1 and 2 in week 3.
CONTINUOUS ASSESSMENT MARKS can be seen on: http://mathsci.ucd.ie/checkmarks (mind that there is a delay of about 2 weeks before your marks are entered by the tutors)

### Homework

Homework is due to hand in during titorials; late assignments are not accepted.
• Homework 1: Plane Geometry (due in week 2)
• Homework 2: Lines and planes in 3-space (due in week 3)
• Homework 3: Coordinate Systems in 3-space, Quadric Surfaces and Their Plane Sections (due in week 4)
• Homework 4: Limits, Continuity and Partial Derivatives (due in week 5)
• Homework 5: Tangent Planes, Directional Derivatives and Gradient Vectors (due in week 6)
• Homework 6: Gradient Curves and Chain Rules for Partial Derivatives (due in week 7)
• Homework 7: Chain Rule, Implicit Functions and Jacobians (due in week 8)
• Homework 8: Partial Derivatives of Higher Orders and Change of Coordinates in Differential Operators (due in week 9)
• Homework 9: Taylor Expansion and Extremum Problems (due in week 11)
• Homework 10: Ordinary Differential Equations and Laplace Transform (due in week 12)

It is very good to read a textbook along with the lecture notes. Practically any book in calculus would work, just check that it contains our topics and read the respective chapters. In James Joyce library one can find these ones on shelf 515:
 [A] Robert A. Adams, Calculus / a complete course [H] Francis B. Hilderbrand, Advanced Calculus for Applications

### Course Outline

1. Plane Geometry
2. Geometry in 3-dimensional Space
• (2.1) Points and Vectors
• (2.2) Planes and Lines
• (2.3) Cylindrical and Spherical Coordinate Systems
• (2.4) Quadric Surfaces and Their Plane Sections
Reading: [A] Chapter 10
3. Derivatives of Functions of Several Variables
• (3.1) Limits and Continuity   (part 2)
• (3.2) Partial and Directional Derivatives
• (3.3) Linear Approximation and Tangent Planes   (part 2)
• (3.4) Gradient Vectors and Gradient Curves   (part 2)
• (3.5) Chain Rule for Partial Derivatives I: Derivative of a Function Given Implicitly
• (3.6) Chain Rule for Partial Derivatives II: Implicit Function Theorem and Jacobians   (part 2)   (part 3)
• (3.7) Partial Derivatives of Higher Order and Change of Coordinates in Differential Operators
• (3.8) Taylor Expansion
Reading: [A] Chapter 12, [H] Chapter 7
4. Extremum Problems
• (4.1) Maxima and Minima of Functions of Several Variables
• (4.2) Extremum Problems
• (4.3) Constrained Extremum and the Method of Lagrange Multipliers
Reading: [A] Chapter 13, [H] Chapter 7
5. Ordinary Differential Equations
• (5.1) Overview
• (5.2) Linear Homogeneous Differential Equations   (part 2)
• (5.3) Non-homogeneous Linear Equations: Variation of Coefficients
• (5.4) The Initial Value Problem and the Laplace Transform   (part 2)   (part 3)
Reading: [A] Chapter 17, [H] Chapters 1 and 2

### Assessment

• 10% continuous assessment (homework)
• 20% midterm exam
• 70% final exam