Auckland Sports College

"Moving Forward Together"

AUT - Engineering Mathematics ll

This is a calculus course which extends differentiation and calculus concepts to three dimensions. It has both calculus and algebra content

Calculus:

• Functions of several variables
• Definition & examples, partial derivatives, chain rule, total differential and applications
• Double integrals, converting double integrals to iterated integrals, double integrals in polar coordinates, applications of double integrals
• Infinite sequences and series (Convergence & divergence of sequences, combination of sequences, partial sums, convergence of series, various tests of convergence of series)
• Taylor series - Power series, Taylor and Maclaurin series, Taylor and Maclaurin series of basic functions (trigonometric, exponential, logarithmic)
• Laplace Transform and applications - Definition, common Laplace Transform, inverse Laplace transform, transform of derivatives, solving ordinary differential equations via Laplace transform

Algebra:

• Vector spaces - Introduction to vector spaces, examples of vectors spaces, linear combinations, linear independence
• Linear transformations - Introduction to linear transforms, examples, linear transforms and the matrix representation, rotational transforms & examples
• Eigenvalues and eigenvectors - Definition, examples & geometric interpretation, bases and diagonalisation of matrices, applications of eigenvalues and eigenvectors

Probability and statistics

• Revision of set operations, mathematical probability, conditional probability, random variables, discrete random variables, mean & variance of a discrete random variable, special distributions, the normal distribution

Week 1 - Functions of Several Variables, Vector Functions, Space Curves

There is some terminology to become familiar with, such as R² the x-y plane and R³. three dimensional space.

We need the idea of a function of two variables. The volume of a cylinder is πr²h which we write as V(r,h) = πr²h. The radius and the height are independent variables - they can have any value. V(r,h) is a dependent variable as its value depends on the values of 'r' and of 'h''

The graph of a function of two variables is three dimensional : for example z = V(r,h) = πr²h

So, the graph of z = f(x,y) = x² is a solid parabola which runs along the length of the y-axis. For any value of y, z = x²

A level curve, also called a contour curve of a surface is the set of all values where f(x,y) = c, for some constant term 'c'. An example of contour curves is the 'height above sea level' curves on a topographical map.

A function is a rule that assigns each value in the domain of the function a single value in the range of the function. A vector-valued function (vector function) is a function whose domain is a set of real numbers and whose range is a set of vectors. This course concentrates on functions where x,y,z are functions of t (most commonly t is time).

So, and this is important, r(t) = f(t)i + g(t)j + h(t)j

We write this as r(t) = <f(t), g(t), h(t)> = f(t)i + g(t)j + h(t)j

A continuous vector function defines a "space curve" that is the tip of the moving vector. The equations f(t), g(t), and h(t) are parametric equations and 't' is the parameter.

r'(t) = <f'(t), g'(t), h'(t)> = f'(t)i + g'(t)j + h'(t)j

Week 2 - Applications of Vector Functions, limits and continuity, partial derivatives

Partial Derivatives The partial derivative of a function of two (or more) variables is the derivative with respect to a single variable. The other variables are treated as constants.

if f(x,y) = x³ + x²y³ - 2y²

then f_x(x,y) = 3x² + 2xy³

and f_y(x,y) = 3x²y² - 4y

The partial derivative is the gradient of the surface in the 'x' or the 'y' direction.

Higher order partial derivatives - f_xx means the second partial derivative with respect to 'x'. f_xy means the partial derivative with respect to x is calculated first, then the partial derivative with respect to y

Week 3 - Chain Rule, Implicit Differentiation, tangent planes, and linear approximations

Week 4 - Directional Derivatives, Maxima, Minima

If we think of a point, any point, on a 3-dimensional surface (for example part-way up a hill) in which direction is the slope a maximum? In which direction is the slope a minimum?

The first thing we need to do is to calculate the slope in an arbitrary direction 'u' where 'u' is a unit vector and u = ai +bj for some a, b. If 'u' is a unit vector then x = icosθ and y = jsinθ where θ is the positive angle formed by 'u' and the x-axis.

The directional derivative (D_u) of a differentiable function f(x,y) in the direction of a unit vector u = < a, b > is

D_uf(x,y) = f_x(x,y)a + f_y(x,y)b

If 'u' makes an angle θ with the positive x-axis then we can write D_u as

D_uf(x,y) = f_x(x,y)cosθ + f_y(x,y)sinθ

The vector < f_x(x,y), f_y(x,y) > is called the gradient of 'f' and is written as ∇f(x,y)

and the formula for the directional derivative can be written as a dot product

D_uf(x,y) = ∇f(x,y).u

The maximum value of the directional derivative D_u is |∇f(x,y)| and it occurs when the vector u has the same direction as the gradient vector (as the dot product of 2 vectors is |a||b|cosθ and cos is a maximum when &theta = 0.

For a point (x,y), ∇f(x,y) is perpendicular to the level curve through (x,y)

Gradient vectors determine the maximum rate of change. They are a different concept to the maximum and minimum values of a function.

Maximum and Minimum values The point (a,b) is a critical point if f_x(a,b) = 0 (or doesn't exist) AND f_y(a,b) = 0 (or doesn't exist).

Local maxima and local minima are critical points, but not all critical points are local maxima or local minima. To determine whether a critical point is a local maxima or a local minima we need to use a second derivative test.

If (a,b) is a critical point of 'f', define

D = D(a,b) = f_xx(a,b)f_yy(a,b) - (f_xy(a,b))²

if D > 0 and f_xx(a,b) > 0, then (a,b) is a local minimum

if D > 0 and f_xx(a,b) < 0, then (a,b) is a local maximum

if D < 0 then (a,b) is a saddle point

To find the global maximum and the global minimum (the extreme values) all the critical points must be evaluated as well as any boundary points.