Edit: What is spec really based on? Is it just extended calculus from methods 1/2 kinda thing. Cause were about to start calculus and i had a look through it and seemed like id enjoy it.
In 2015, you'll have 5 areas of study - since you do physics, I'm just not going to explain a few the things in here and assume you know what the words I use mean. If you don't know what they mean, ask me and I'll explain, or look them up online.
AoS1 - Graphs and relations.
Methods does not cover everything to with relations and their graphs (in fact, it barely covers anything...). In specialist, you'll learn about parametric equations (relations of the form x=cos(t) and y=sin(t), where the x and y values are based on the "parameter" t), graphs with oblique asymptotes (basically, an asymptote which is a curve, not just x=h or y=k) and reciprocal graphs (of the form 1/f(x)).
On top of this, there's a much bigger focus on circular functions than there is in methods, including inverse circular functions and the reciprocal circular functions.
AoS2 - Algebra
It's stupid that they call this algebra - minus partial fractions (which I'll talk about in AoS3), the whole area of study is based on complex numbers.
Basically, you'll be aware of real numbers, but beyond the reals, you have a bunch of numbers that aren't real. One example of this is the imaginary number, i. i is defined such that
, which means that i itself is the square root of a negative number (something that "can't be done"). This gives rise to a type of numbers called "complex numbers", which are of the form a + bi. In this area of study, you'll learn about graphing these numbers, converting between a few different ways of writing them, and sketching their sets in a modified cartesian plane called an "Argand Plane". Complex numbers are different in that they use a polar co-ordinate system, as opposed to a rectangular co-ordinate system that you're used to. Don't worry too much about that right now, just know that complex numbers are pretty cool and are really different to what you know of with the "regular" kind of numbers.
AoS3 - Calculus
Quite possibly the largest area of study in the course - not because it's content heavy, but because it has implications for nearly everything in the course (however, less implications than methods has). Comes in three components - differentiation, integration and differential equations.
Differentiation is almost no different to what you learn in methods, however there is added in there implict differentiation (differentiation non-functions) and double derivatives (although they're basically a wasted concept in specialist...), as well as derivatives of the new circular stuff you learned.
Integration is the big stuff - in methods, you just have to know basic integrals, such as the integral of cos(x) is sin(x). In specialist, they kick it up massively, and you have to know the integral of all the derivatives you've learned thus far AS WELL AS learning integral techniques to solve things such as
, something that no methods student can do. These integral techniques include partial fractions, scalar modifications (fancy name I came up with just then, because I don't think there is a real name for them
) and substitution.
Differential Equations is the combination of the two - giving you an equation of the form
)
and asking you to either show that a given solution is true (the solution being finding an equation of the form
)
), to solving the actual equation itself. Also covers slope fields and euler's method, which are numerical approximations of differential equations.
Also in differential equations is a large study of kinematics - basically what you know from physics, but using calculus. This was my favourite part of specialist personally, and when I got to non-calc physics at uni, I cried and hated it.
AoS4 - Vectors
This area of study is a proper, mathematical study of vectors (yes - maths had them first, not physics. This qualifies for most of the things in physics, btw - it was in maths waaaaaay before it was in physics.
). You'll learn what a vector is, how to find its length, the angle it makes with particular axis, how to project one vector onto another vector, finding perpendicular and parallel vectors, how to determine if a vector is perpendicular or parallel to one another, finding the angles between two vectors, how to write one vector in the terms of two others - and if this can always be done or not (spoiler: it can't always). Then, you move on to vector functions, which is basically just parametric equations rehashed. Often combined with kinematics, where "this vector function shows the displacement of a particle at time t. Find its acceleration at t=5".
AoS5 - Mechanics
Basically, easy mechanics that you know from physics. Really easy. Only difference is that this area of study is done with a proper treatment of vectors, so everything can be done properly, particularly curvillinear motion. Otherwise, though... Not really much to say here - it's just Newtonian Mechanics, I don't know what I could say about it...
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