Mathematics, second course

Aim

The aim of the course is to give the students basic proficiency in the single- and multivariable calculus needed for their further studies in chemistry, especially physical chemistry. After fulfilling the course the student should be able to perform elementary calculations in the areas specified below. Thus, the student should be able to

• Use standard techniques to calculate antiderivatives and definite integrals.
• Express and calculate geometrical quantities like areas of plane regions, arc length, volumes of solids of revolution and areas of solids of revolution.
• Handle first-order separable and first-order linear differential equations and integrale quations.
• Explain Taylor’s formula
• Use Taylorexpansions to approximate functions and investigate limits.
• calculate partial derivatives of elemtary functions and compositions of these in several variables
• calculate the differential of a function and use it to estimate the error propagation in an approximation
• calculate extreme values of functions definied on restricted domains of simple geometry
• calculate double integrals over triangular and rectangular domains
• calculate double integrals over circle sectors by using polar coordinates
• calculate triple integrals over domains shaped as a parallelepiped when represented in either cartesian or spherical coordiantes

Content

Primitive functions. Integration and geometrical applications, including area, curve length, areas of rotation and volumes of rotation. Improper integrals. Taylor's formula. Maclaurin expansions of elementary functions with applications to the calculation of limits. Linear ordinary differential equations of first and second order, separable equations. Functions of several variables, partial derivatives, the chain rule and error propagation. Gradient, tangents and tangentplanes. Extreme values. Double and triple integrals.