By Russell L. Herman
Advent and ReviewWhat Do i have to comprehend From Calculus?What i want From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe easy Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical ideas of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali. Read more...
summary: advent and ReviewWhat Do i must understand From Calculus?What i want From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe easy Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical recommendations of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali
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2 Trigonometric Functions Another set of useful functions are the trigonometric functions. These functions have probably plagued you since high school. They have their origins as far back as the building of the pyramids. Typical applications in your introductory math classes probably have included finding the heights of trees, flag poles, or buildings. It was recognized a long time ago that similar right triangles have fixed ratios of any pair of sides of the two similar triangles. These ratios only change when the non-right angles change.
The Partition Function A common occurrence of geometric series is a series of exponentials. An example of this occurs in statistical mechanics. Statistical mechanics is the branch of physics which explores the thermodynamic behavior of systems containing a large number of particles. An important tool is the partition function, Z. This function is the sum of terms, e− n /kT , over all possible quantum states of the system. Here, n is the energy of the nth state, T the temperature, and k is Boltzmann’s constant.
Differentiate H ( x ) = 5 cos π tanh 2x2 . This is a composition of three functions: H ( x ) = f ( g(h( x ))), where f ( x ) = 5 cos x, g( x ) = π tanh x, and h( x ) = 2x2 . Then the derivative becomes H (x) = 5 − sin π tanh 2x2 d dx π tanh 2x2 = −5π sin π tanh 2x2 sech2 2x2 d 2x2 dx = −20πx sin π tanh 2x2 sech2 2x2 . 56) Integrals Integration is typically a bit harder. 56) and having to figure out what was differentiated in order to get the given function. As you may recall from the Fundamental Theorem of Calculus, the integral is the inverse operation to differentiation: df dx = f ( x ) + C.