The language of physics
is mathematics. In order to study physics seriously, one needs to learn
mathematics that took generations of brilliant people centuries to work
out. Algebra, for example, was cuttingedge mathematics when it was being
developed in Baghdad in the 9th century. But today it's just the first
step along the journey. 
Algebra 

Algebra provides the first exposure
to the use of variables and constants, and experience manipulating and
solving linear equations of the form y = ax + b and quadratic equations
of the form y = ax^{2}+bx+c. 
Geometry 

Geometry at this level is twodimensional
Euclidean geometry, Courses focus on learning to reason geometrically,
to use concepts like symmetry, similarity and congruence, to understand
the properties of geometric shapes in a flat, twodimensional space. 
Trigonometry 

Trigonometry begins with the study
of right triangles and the Pythagorean theorem. The trigonometric functions
sin, cos, tan and their inverses are introduced and clever identities
between them are explored. 
Calculus (single
variable) 

Calculus begins with the definition
of an abstract functions of a single variable, and introduces the ordinary
derivative of that function as the tangent to that curve at a given point
along the curve. Integration is derived from looking at the area under
a curve,which is then shown to be the inverse of differentiation. 
Calculus (multivariable) 

Multivariable calculus introduces
functions of several variables f(x,y,z...), and students learn to take
partial and total derivatives. The ideas of directional derivative, integration
along a path and integration over a surface are developed in two and three
dimensional Euclidean space. 
Analytic Geometry 

Analytic geometry is the marriage
of algebra with geometry. Geometric objects such as conic sections, planes
and spheres are studied by the means of algebraic equations. Vectors in
Cartesian, polar and spherical coordinates are introduced. 
Linear Algebra 

In linear algebra, students learn
to solve systems of linear equations of the form a_{i1} x_{1}
+ a_{i2} x_{2} + ... + a_{in} x_{n} =
c_{i} and express them in terms of matrices and vectors. The properties
of abstract matrices, such as inverse, determinant, characteristic equation,
and of certain types of matrices, such as symmetric, antisymmetric, unitary
or Hermitian, are explored. 
Ordinary Differential
Equations 

This is where the physics begins!
Much of physics is about deriving and solving differential equations.
The most important differential equation to learn, and the one most studied
in undergraduate physics, is the harmonic oscillator equation, ax'' +
bx' + cx = f(t), where x' means the time derivative of x(t). 
Partial Differential
Equations 

For doing physics in more than one
dimension, it becomes necessary to use partial derivatives and hence partial
differential equations. The first partial differential equations students
learn are the linear, separable ones that were derived and solved in the
18th and 19th centuries by people like Laplace, Green, Fourier, Legendre,
and Bessel. 
Methods of approximation 

Most of the problems in physics can't
be solved exactly in closed form. Therefore we have to learn technology
for making clever approximations, such as power series expansions, saddle
point integration, and small (or large) perturbations. 
Probability and
statistics 

Probability became of major importance
in physics when quantum mechanics entered the scene. A course on probability
begins by studying coin flips, and the counting of distinguishable vs.
indistinguishable objects. The concepts of mean and variance are developed
and applied in the cases of Poisson and Gaussian statistics. 