618-181 Mathematics 1R

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Objectives:

On completion of this subject students should:

Comprehend:

• the manipulation of vectors, matrices, and systems of linear equation;

• the concepts of solid geometry;

• the properties of basic functions of calculus;

• the classification and principles governing the solution of the basic first and second order differential equations;

• the range of calculus skills and techniques necessary for the solution of these differential equations, and the solution methods applicable to each type.

Have developed:

• the skills required to solve systems of linear equations;

• the skills to employ vector methods in geometrical problems;

• the skills required to differentiate and integrate the basic functions of calculus;

• an ability to use differential calculus to solve extremal problems;

• an ability to compute a wide range of integrals;

• an ability to use integration to compute area, length and volume;

• the ability to classify and solve the basic differential equations of first and second order, and the integral and differential calculus skills to achieve these solutions with accuracy and confidence.

Appreciate:

• the fundamental concepts in linear algebra and calculus necessary for further serious studies in mathematics;

• the role of differential equations in applied mathematics.

Content:

Vectors and matrices: Vector in three-dimensional space, dot and cross products, triple products, determinants; equations of lines and planes, geometrical applications; matrices, row operations, inverses; solution of linear equations, row-reduction, rank. Calculus: functions of one real variable, derivatives; curve sketching; maxima and minima, curvature; antiderivatives and the definite integral; trigonometric functions and their inverses; applications of integration, areas, arc length, surface areas and volumes of solids of revolution. Differential equations: gradient fields, first-order differential equations (linear via integrating factors, separable and homogeneous); linear differential equations with constant coefficients, particular integrals and complementary functions; applications to damped oscillators and resonance.

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