COURSE CONTENT

Cartesian, cylindrical and spherical coordinates in space.  Second degree surfaces.

Multivariable functions, limit, continuity, partial derivative of first or higher order and geometric interpretation. Derivation rules, schwarz theorem. Total differential and the concept of differentiation. Derivatives of composite and implicit functions, implicit function theorem. Jacobian determinant and functional  dependence. Taylor and Maclaurin theorems. Extrema of multivariable functions and constrained extrema, Lagrange multipliers.

Vectors and analytic geometry in space. Limit, continuity and derivative of vector-valued functions. Elements of the differential geometry of curves in space. Position vector of particle, vector velocity and acceleration. Unit tangent, normal and binormal vectors, curvature and torsion of curve. Directional derivative, gradient scalar functions, divergence and rotation of vector functions, their physical interpretation and basic vector identities. Laplace differential operator, harmonic functions and partial differential equations of Helmholtz, wave and diffusion. Potential functions, conservative and solenoidal fields, Helmholtz decomposition theorem. Curvilinear coordinate systems, transformations and change of coordinates. Geometrical applications of partial derivatives.

Multiple integrals, change of coordinate system. Volume of three–dimensional domains, masses and moments in three dimensions. Line integrals, work, circulation and flux. Surface area, surface integral and parameterized surfaces. Gauss, Stokes and Green theorems, physical interpretation.

LEARNING OUTCOMES

To give the student in mechanical engineering the knowledge of applied engineering mathematics that he/she needs in his/her science in the areas of differential/integral calculus and vector analysis. This knowledge is necessary and is used in many subsequent specialization courses in mechanical engineering.