## MA22S1: Multivariable calculus for science

### lecturer: Masha Vlasenko

The prerequisite to this course is single variable calculus MA11S2. 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. By the end of the course you will know how to differentiate and integrate functions of several variables. As the Newton-Leibniz rule relates derivatives to integrals in single variable calculus, in multivariable calculus this is done by the three major theorems (Green's, Stokes' and Gauss'). These are considered in MA22S2 in the second term.

### Syllabus

• Vector-Valued Functions and Space Curves
• Polar, Cylindrical and Spherical Coordinates
• Quadric Surfaces and Their Plane Sections
• Functions of Several Variables, Partial Derivatives
• Tangent Planes and Linear Approximations
• Directional Derivatives and the Gradient Vector
• Maxima and Minima, Lagrange Multipliers
• Double Integrals Over Rectangles and over General Regions
• Double Integrals in Polar Coordinates
• Triple Integrals in Cylindrical and Spherical Coordinates
• Change of Variables, Jacobians

ATTENTION: in contrast to the last year's exam paper, there will be no formulas and hints given this year. Please make sure you remember main formulas given in class.
A collection of useful formulas and facts from the lectures

Scholarship exam (January 9th 2013) questions and solutions.
Niall Redmond solved questions 1,2 and 4. Question 1 was also solved by Michael Tierney, and there were good though incomplete solutions by Conor Byrne and Jacub Jadwiszczak. Question 2 was also solved by Darragh Keane and Lua Koenig, and there were several partial solutions too. In Question 4 the best results were obtained again by Michael Tierney and Darragh Keane, though many students computed either the volume or the surface area. Question 3 appeared to be the hardest, the highest grades there were 20%. I think the exam was difficult, and these are really good results. My regards, and all the best!

Calculus. Late transcendentals. by H. Anton, I. Bivens, S. Davis.

### Homework

The tutor is James Boland.
Every student is assigned to one of the three groups A, B or C. The times for tutorials are
(A) Wednesday, 14:00-15:00 in MGLT
(B) Tuesday, 11:00-12:00 in M17
(C) Monday, 9:00-10:00 in Syn2
Homeworks are due to hand in during the corresponding classes; late assignments are not accepted. Marking scheme: each exercise is marked with 1 point. Exercises with (*) are optional, they are not counted for your final mark. Though it is recommended to try them, it might be also helpful to prepare for the scholarship exam. Exercises with (**) are for curious people only, the matherial will not appear in any of the exam papers.

### Lectures (extended syllabus)

#### Vector-valued functions and space curves

1. Parametric plane curves.
2. Parabola, ellipse and hyperbola.
3. Shifts and rotations of rectangular coordinate system.
5. Planes and lines in 3-space. Distance problems.
6. Quadric surfaces and their plane sections.
7. Parametric space curves.
8. Cylindrical and spherical coordinates in 3-space.
9. Differentiation and integration of vector-valued functions.
10. Length of a portion of a curve.
11. Change of the parameter. Parametrization by the length.
12. Unit tangent, normal and binormal vectors.
13. Curvature.
14. (*) Kepler's Laws of Planetary Motion.

#### Functions of several variables

1. Limits and continuity.
2. Partial and directional derivatives.
3. Linear approximation and planes tangent to the graph.
4. Partial derivatives of higher order.
6. Chain rule for partial derivatives.
7. Gradient vectors and level curves.
8. Chain rule and change of coordinates in differential operators.
9. Maxima and minima of functions of two variables.
10. Extremum problems.
11. Finding global extrema over a closed and bounded set.
12. Method of least squares.
13. Constrained extremum principle: the method of Lagrange multipliers.

#### Multiple integrals

1. Double integrals as iterated integrals. Fubini's theorem.
2. Double integrals in polar coordinates.
3. Using double integrals to compute volumes of solids.
4. Area of a portion of a surface.
5. Triple integrals in rectangular, cylindrical and spherical coordinates.
6. Change of variables in multiple integrals. Jacobians.
7. Applications: averages, center of gravity and centroid.

### Assessment

2 hour examination in May, the final mark will be 80% of the exam mark + 20% of the home assignments mark

### Learning outcome

On sucessful completion of this module, students will be able to:
• write equations of planes, lines and quadric surfaces in the 3-space
• determine the type of conic section and write change of coordinates turning a quadratic equation into its standard form
• use cylindrical and spherical coordinate systems
• write equations of a tangent line, compute unit tangent, normal and binormal vectors and curvature at a given point on a parametric curve; compute the length of a portion of a curve
• apply above concepts to describe motion of a particle in the space
• calculate limits and partial derivatives of functions of several variables
• write local linear and quadratic approximations of a function of several variables, write equation of the plane tangent to its graph at a given point
• compute directional derivatives and determine the direction of maximal growth of a function using its gradient vector
• use the method of Lagrange multipliers to find local maxima and minima of a function
• compute double and triple integrals by application of Fubini's theorem or use change of variables
• use integrals to find quantities defined via integration in a number of contexts (such as average, area, volume, mass)