References

Course Outline

Pre-requisite: Foundation Mathematics

1.

Analytic Geometry The general equation \(s(x,y) = 0\) of the 2nd degree and its reduction to conical form; classification of conic sections; the significant properties of the conic sections; the \((r,q)\) equations for the parabola, ellipse and hyperbola.

2.

Differential Calculus of Functions of One Variable Tangents and normals to general plane curves; Rolles Theorem; the mean value theorem and its generalization to Taylor’s theorem; Application of Taylor’s theorem; Power series for \(e^x\), \(\sin x \), \(\cos x\), \(\log (1+x)\) etc; Limits and limit rules (L’Hospital); Curvature, intrinsic coordinates and the transformation \((x,y) - (s,u)\).

3.

Integration Calculus of Functions of One Variable Further methods of indefinite integration (e.g transformation, integration by parts, recurrence formula, etc); The definite integral as the limit of a sum (treated heuristically) with application to areas; volumes; length of curves; centroids; moments of inertia.

4.

Vector Analysis Theory of geometry vectors with applications: Vector and parametric equations; differentiation of vectors; the notion of a projectile; curvature; the unit tangent and normal vectors; arc length as a parameter etc; Application to 3-dimensional spaces: Vectors in three dimensional spaces; equations of straight lines in space; the scalar and vector products of two vectors; computations with vector products; equations of planes, spaces curves etc.

5.

Differential Calculus of Functions of Several Variables Functions of several variables; partial derivatives of various orders and their manipulation; The total differential; chain rules for partial and total differentiation; application of the total differential to error estimation; Stationary points, etc; Euler’s theorem for homogeneous functions.

6.

Ordinary Differential Equations Order, degree and general solutions of D.E; Linear and non-linear equations; The general linear equation; (a) homogeneous, (b) non-homogeneous; properties of the linear homogeneous D.E. (eg. super-position of solutions); Properties of the linear non-homogeneous D.E. its general solution (GS), complementary function (CF) and particular integral (PI),
GS = CF + PI; The solvable D.E’s (Linear and non linear) of O(1), their classification into exact, variable separable, homogeneous, linear, Bernoulli types etc, and methods for their solution; The linear equations of O(2) with constant or homogeneous coefficients and their methods of solutions; The methods of variation of parameters and solution by series.