Collection of Problems in Higher Mathematics

[6314/73%]

N. M. Gunter, R. O. Kuzmin

Chapter 1Analytic Geometry in the Plane[383/76%]

Section 1.1Vectors, Projections, and Coordinates in the Plane. Simplest Applications[38/84%]

Section 1.2The Line and the Circle[91/86%]

Section 1.3Loci[34/76%]

Section 1.4Second-Order Curves in Simplest Form[90/74%]

Section 1.5Second-Order Curves Given by an Equation in General Form[17/71%]

Section 1.6Center, Diameters. Simplification of Equations of Second-Order Curves[33/67%]

Section 1.7Conjugate Diameters. Axes of Symmetry. Asymptotes[15/80%]

Section 1.8Foci and Directrices[18/94%]

Section 1.9Tangents to Second-Order Curves. Poles and Polars[13/54%]

Section 1.10Miscellaneous Problems[34/53%]

Chapter 2Analytic Geometry in Space[333/76%]

Section 2.1Vectors and Coordinates in Space[39/67%]

Section 2.2The Plane[36/89%]

Section 2.3The Line in Space[66/77%]

Section 2.4Generation of Surfaces[30/90%]

Section 2.5Second-Order Surfaces. Center and Diametral Planes[39/74%]

Section 2.6Tangent Planes and Lines to Second-Order Surfaces[42/43%]

Section 2.7Simplification of Equations of Second-Order Surfaces[41/78%]

Section 2.8Circular Sections, Rectilinear Generators, and Other Problems[40/95%]

Chapter 3Differential Calculus[668/69%]

Section 3.1Theory of Limits[116/100%]

Section 3.2Miscellaneous Problems[37/14%]

Section 3.3The Concept of a Function. Continuity. Graphical Representation of Functions[98/26%]

Section 3.4Finding Derivatives[117/97%]

Section 3.5The Geometric Meaning of the Derivative[29/66%]

Section 3.6Higher-Order Derivatives[85/62%]

Section 3.7Functions of Several Variables. Their Derivatives and Differentials[102/60%]

Section 3.8Differentiation of Implicit Functions[39/79%]

Section 3.9Change of Variables[45/82%]

Chapter 4Applications of Differential Calculus to Analysis[489/57%]

Section 4.1The Theorems of Rolle, Lagrange, and Cauchy. Increase and Decrease of Functions. Inequalities[36/6%]

Section 4.2Finding the Greatest and Least Values of Functions of One Variable[43/95%]

Section 4.3Plotting Graphs of Functions[57/47%]

Section 4.4Miscellaneous Problems on Greatest and Least Values[64/98%]

Section 4.5Series, Their Convergence[61/59%]

Section 4.6Expansion into Series[43/0%]

Section 4.7Series and Operations on Them[57/0%]

Section 4.8Evaluation of Indeterminate Forms[39/64%]

Section 4.9Extremal Values of Functions of Several Variables[89/94%]

Chapter 5Geometric Applications of Differential Calculus[534/73%]

Section 5.1Equations of Curves and Their Types[41/68%]

Section 5.2The Tangent and the Normal[90/54%]

Section 5.3Convexity, Curvature, and Radius of Curvature[53/68%]

Section 5.4Evolutes of Curves[23/96%]

Section 5.5Envelopes of Curves[23/96%]

Section 5.6Plotting Curves[124/96%]

Section 5.7Space Curves: The Tangent Line and the Normal Plane[39/56%]

Section 5.8Space Curves: The Osculating Plane, the Principal Normal, and the Binormal[44/80%]

Section 5.9Surfaces. Their Equations[24/83%]

Section 5.10Tangent Planes and Normals. Envelopes[48/54%]

Section 5.11Lines on Surfaces and Curvature of Surfaces[25/52%]

Chapter 6Higher Algebra[389/73%]

Section 6.1Complex Numbers[76/87%]

Section 6.2Factoring a Polynomial. The Relation Between Coefficients and Roots[40/40%]

Section 6.3A Polynomial with Real Coefficients. Rolle's Theorem[39/15%]

Section 6.4Rational Fractions. Decomposition into Partial Fractions[25/92%]

Section 6.5Determinants. Systems of Linear Equations[50/72%]

Section 6.6Matrices. The Characteristic Equation. Quadratic Forms[28/64%]

Section 6.7Symmetric Functions[33/82%]

Section 6.8Transformation and Solution of Equations[40/88%]

Section 6.9Isolation and Computation of Roots[58/98%]

Chapter 7Integration of Functions[460/99%]

Section 7.1Basic Formulas and Techniques of Integration[102/100%]

Section 7.2Integration of Rational Functions[76/97%]

Section 7.3Integration of Irrational Functions[80/100%]

Section 7.4Integration of Transcendental Functions[108/97%]

Section 7.5Miscellaneous Problems[94/100%]

Chapter 8Applications of Integral Calculus[206/90%]

Section 8.1The Definite Integral[49/82%]

Section 8.2Computing Areas (Quadrature of Curves)[60/88%]

Section 8.3Computing Arc Lengths of Curves[32/88%]

Section 8.4Computing Volumes[37/97%]

Section 8.5Computing Areas of Surfaces of Revolution[28/100%]

Chapter 9Multiple Integrals and Line and Surface Integrals[365/85%]

Section 9.1Introduction[34/100%]

Section 9.2Computing Areas[36/94%]

Section 9.3Computing Volumes[56/93%]

Section 9.4Computing Surfaces[28/100%]

Section 9.5Line Integrals[17/94%]

Section 9.6Some Applications of Double Integrals in Mechanics and Strength of Materials[27/89%]

Section 9.7Surface Integrals, Coordinates of Centers of Gravity, and Moments of Inertia of Surfaces[14/93%]

Section 9.8The Triple Integral[7/100%]

Section 9.9Computing Volumes[40/100%]

Section 9.10Coordinates of Centers of Gravity and Moments of Inertia of Solids[37/86%]

Section 9.11Integrals of Field Theory and Potential Theory[57/39%]

Section 9.12Multiple (Iterated) Integrals[12/58%]

Chapter 10Differential Equations[713/95%]

Section 10.1Forming Differential Equations from Their Given Integrals[26/92%]

Section 10.2Finding Functions from Their Total Differential[13/92%]

Section 10.3Integration of Total Differentials[7/100%]

Section 10.4Equations with Separable Variables[51/92%]

Section 10.5Homogeneous Equations and Those Reducible to Them[31/94%]

Section 10.6Linear Equations and Those Reducible to Them[34/97%]

Section 10.7The Riccati Equation[12/100%]

Section 10.8The Jacobi Equation[7/100%]

Section 10.9The Integrating Factor[22/91%]

Section 10.10Euler Equations[14/29%]

Section 10.11Equations Not Solved for y'[32/100%]

Section 10.12Singular Solutions of Equations[10/100%]

Section 10.13Problems on Trajectories[34/100%]

Section 10.14Miscellaneous Problems[81/98%]

Section 10.15Higher-Order Equations Admitting a Reduction of Order[86/94%]

Section 10.16Linear Equations with Constant Coefficients and Equations Reducible to Them[91/100%]

Section 10.17Linear Equations. Miscellaneous Problems[38/95%]

Section 10.18Systems of Differential Equations[54/94%]

Section 10.19First-Order Linear Partial Differential Equations[70/99%]

Chapter 11Definite Integrals[414/57%]

Section 11.1The Definite Integral as the Limit of a Sum[15/47%]

Section 11.2Mean Value Theorems. Improper Integrals[42/38%]

Section 11.3Computing Definite Integrals by Integration and Substitution[26/8%]

Section 11.4Finding Integrals by Means of Reduction Formulas[20/90%]

Section 11.5Integration by Means of Series[44/75%]

Section 11.6Differentiation and Integration Under the Integral Sign[32/88%]

Section 11.7Euler Integrals[41/76%]

Section 11.8Miscellaneous Problems[96/53%]

Section 11.9Fourier Series and Related Questions[98/52%]

Chapter 12Series[233/22%]

Section 12.1Investigation of the Convergence of Series[48/67%]

Section 12.2Direct Summation of Finite Sums and Infinite Series[78/8%]

Section 12.3Finding Sums of Series by Differentiation; Some Series Expansions[45/22%]

Section 12.4Miscellaneous Problems[62/5%]

Chapter 13Approximate Computations[183/55%]

Section 13.1Interpolation. Error Theory[35/94%]

Section 13.2Approximate Computation of Integrals[28/64%]

Section 13.3The Euler--Maclaurin Formula and Similar Methods[18/17%]

Section 13.4Acceleration of Convergence of Series[36/17%]

Section 13.5Computing Integrals by Means of Series[25/44%]

Section 13.6Numerical Solution of Equations[12/100%]

Section 13.7Approximate Integration of Differential Equations[29/62%]

Chapter 14Functions of a Complex Variable[426/43%]

Section 14.1The Cauchy--Riemann Equations[16/75%]

Section 14.2Singular Points of a Function. Taylor and Maclaurin Series[36/11%]

Section 14.3Residues and Their Applications[66/42%]

Section 14.4Distribution of the Zeros of a Function[23/26%]

Section 14.5Expansion of Functions into Partial Fractions[30/0%]

Section 14.6Other Series Expansions[36/53%]

Section 14.7Generating Functions and Special Polynomials[11/0%]

Section 14.8Conformal Mappings[56/73%]

Section 14.9The Maximum Modulus Principle[29/0%]

Section 14.10Differential Equations in a Complex Variable[46/46%]

Section 14.11Applications to Problems of Mathematical Physics[77/69%]

Chapter 15Equations of Mathematical Physics[164/65%]

Section 15.1Forming Second-Order Partial Differential Equations[20/0%]

Section 15.2Reduction of Second-Order Linear Equations to Canonical Form[7/100%]

Section 15.3The Method of Characteristics[17/24%]

Section 15.4The Riemann Method[5/40%]

Section 15.5The Fourier Method[54/83%]

Section 15.6Integral Equations[61/80%]

Chapter 16Calculus of Variations[177/88%]

Section 16.1The Euler--Lagrange Equations[15/93%]

Section 16.2Necessary and Sufficient Conditions for the Simplest Problem of the Calculus of Variations[22/95%]

Section 16.3Parametric Form of Integrals, Transversality[13/92%]

Section 16.4The Hamilton--Jacobi Equation[22/82%]

Section 16.5Integrals Depending on Higher-Order Derivatives or on Several Functions[28/75%]

Section 16.6Discontinuous Solutions. One-Sided Variation[18/94%]

Section 16.7[NO TITLE][21/81%]

Section 16.8Isoperimetric Problems[28/93%]

Section 16.9Miscellaneous Problems[10/100%]

Chapter 17Probability Theory[177/95%]

Section 17.1Application of the Basic Theorems. Bayes' Formula[50/94%]

Section 17.2Mathematical Expectations. The Method of Finite Differences and Generating Functions[34/97%]

Section 17.3Bernoulli's Theorem. Chebyshev's Inequalities[13/100%]

Section 17.4The Theorems of Laplace and Lyapunov--Markov[22/100%]

Section 17.5Geometric Probabilities and Probability Distribution Laws[58/93%]