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The concepts behind each simulator, explained as in class —with figures, tables and formulas— and worked examples that you then solve in the simulator with one click.
Math
Functions and their graphsDomain and range, transformations of a graph, asymptotes of rational functions, inverse functions and implicit, parametric and polar curves.Equations, inequalities and systemsLinear and quadratic equations, the discriminant, factoring, fractions and roots (extraneous solutions), the sign chart and linear systems by Gaussian elimination.Limits and continuityThe idea of a limit, one-sided limits, continuity, indeterminate forms (factoring, conjugate, L'Hôpital), notable limits and limits at infinity.Derivatives and their applicationsDefinition as a limit, product, quotient and chain rules, tangent line, increase and decrease, maxima and minima and optimization.Integrals: antiderivatives, areas and methodsAntiderivatives, the definite integral as a sum of rectangles, the fundamental theorem, area between curves, improper integrals and methods: substitution, by parts and partial fractions.Ordinary differential equationsGeneral and particular solutions, slope fields, separable and linear first-order equations, second order with constant coefficients, undetermined coefficients and initial conditions.Matrices and linear algebraMatrix product, determinant, inverse, rank and row reduction, eigenvalues and eigenvectors and diagonalization.Series and Taylor polynomialsPartial sums, geometric and p-series, ratio, root, comparison, integral and Leibniz tests, absolute and conditional convergence, and Taylor and Maclaurin polynomials.Geometry: areas, volumes and trianglesAreas and perimeters of plane figures, volumes and surface areas of solids, solving for a measurement, the laws of sines and cosines and the ambiguous case.Descriptive statisticsMean, median and mode, sample and population variance and standard deviation, quartiles, box plots, outliers, skewness and frequencies.
Chemistry
The periodic table and electron configurationGroups, periods and blocks, Madelung's rule and its exceptions, effective nuclear charge and trends in radius, ionization energy and electronegativity.Moles, balancing and stoichiometryMoles and molar mass, balancing by trial and error, algebra and matrices, redox and ionic equations, limiting reactant, excess, yield and gases.Solutions: concentration, preparation and ionic strengthMolarity, molality, percentages, ppm and mole fraction; density and basis of calculation; preparing and diluting; ionic strength and activity coefficients.Acids, bases, pH and titrationThe pH scale, strong and weak acids and bases, Ka and Kb, salts, buffers and Henderson–Hasselbalch, titration curves, indicators and polyprotic acids.Spectrophotometry: the Beer–Lambert lawTransmittance and absorbance, A = εbc, calibration curves and standards, deviations from the law, mixtures of several species and least squares.
Engineering
Beams: reactions, shear and momentSupports and loads, equilibrium, shear and bending moment diagrams, and deflection.Beam stresses and failure criteriaNormal and shear stress in the section, principal stresses, von Mises and Tresca, criteria for brittle materials and sizing the section.Buckling of compression membersEuler critical load, effective length, slenderness, the Johnson formula and the buckling safety factor.Mohr's circle and stress transformationPlane stress, transformation equations, principal stresses, maximum shear, building the circle and the absolute maximum shear.Columns: two axes, AISC and eccentric loadsBuckling about each axis and bracing, steel column strength per AISC 360 (LRFD and ASD), the secant formula and higher modes.Statics and trussesTwo-force members, determinacy, method of joints, zero-force members, method of sections and space trusses.Structural analysis: the stiffness methodDegree of indeterminacy, member stiffness matrix, assembly, fixed-end forces, axial deformation, torsion and 2D and 3D frames.Transmission shafts and fatigueLoads in two planes, alternating stresses, the Marin endurance limit, stress raisers (Kt, q, Kf), Goodman, Gerber, ASME and Soderberg, deflection and critical speed.Planar mechanisms: degrees of freedom and kinematicsGrübler–Kutzbach, the Grashof condition, the vector loop, velocity and acceleration of the four-bar, the transmission angle and the slider-crank.Robot arm: forward and inverse kinematicsJoints and degrees of freedom, Denavit–Hartenberg parameters, forward and inverse kinematics, the Jacobian, singularities and trajectories.Gears and gear trainsLaw of gearing and the involute, module and tooth geometry, contact ratio, simple, compound and planetary trains, forces, Lewis and worm drives.Direct-current electric circuitsOhm's law, Kirchhoff's laws, series and parallel, the Δ–Y transformation, superposition, nodal analysis, Thévenin and RC and RL transients.Electric charges, field and Gauss's lawCoulomb's law and superposition, electric field, potential and energy, continuous distributions and Gauss's law with its three symmetries.Digital electronicsNumber systems, two's complement and IEEE 754, Boolean algebra, Karnaugh maps and Quine–McCluskey, combinational circuits, flip-flops and state machines.Control systems: blocks, response and stabilityOpen and closed loop, transfer function, poles and zeros, block diagram algebra, first- and second-order response, Routh–Hurwitz and steady-state error.Control: root locus, Bode and designRoot locus rules, Bode diagram, phase and gain margins, specifications in the s-plane, PID with Ziegler–Nichols and lead and lag compensators.Thermodynamic properties and using the tablesPure substance and phases, the saturation dome, quality, the region of a state, single and double interpolation, compressed liquid and the ideal gas with Pr.Multi-domain simulation: how the simulator computesAnalogies between domains, nodal analysis with states, BDF2 integration with Newton, step size, linearization, poles and G(s), and how to check the results.