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differential calculus

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Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005453
EISBN: 978-1-62708-196-2
... Abstract This article describes the models of physical phenomena involving solution of differential equations such as ordinary or partial differential equations. It reviews the basic concepts of differential calculus and tabulates the expansion of functions into power series. A table of linear...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005451
EISBN: 978-1-62708-196-2
... transforms. laplace transformations THE METHODS OF LAPLACE AND FOURIER TRANSFORMATION and the Heaviside operational calculus are essentially aspects of the same method. This method simplifies solutions of such problems as ordinary differential equations with constant coefficients, linear...
Book Chapter

Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005452
EISBN: 978-1-62708-196-2
... Abstract This article is a comprehensive collection of formulas, tables, and theorems associated with integral calculus. It also summarizes the applications of single, double, and triple integrations in various fields. integral calculus definite integrals line integrals surface...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002446
EISBN: 978-1-62708-194-8
... of the weighted objective method by minimizing the difference between each individual objective and its respective target value in a least-squares sense ( Ref 8 ). Optimization Algorithms When F is an algebraic function of the design variables, classical methods from elementary calculus can be used...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005233
EISBN: 978-1-62708-187-0
... to solve the CFD equations are described, including geometry description and discretization. Finally, the application of CFD to a few casting problems is demonstrated. Fundamentals of CFD Computational fluid dynamics has as its objective the numerical solution of fluid-flow equations. The calculus...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005426
EISBN: 978-1-62708-196-2
... and process design Background and History Computational fluid dynamics has as its objective the numerical solution of fluid-flow equations. The calculus problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002444
EISBN: 978-1-62708-194-8
... design Computational fluid dynamics has as its objective the numerical solution of fluid-flow equations. The calculus problem of solving a coupled system of nonlinear partial differential equations (PDEs) for the variables of interest (e.g., velocity, pressure, and temperature) is transformed...
Series: ASM Handbook
Volume: 4B
Publisher: ASM International
Published: 30 September 2014
DOI: 10.31399/asm.hb.v04b.a0005923
EISBN: 978-1-62708-166-5
... production of parts through a continuous carburizer. Plants involved in heat treating rear and front axle assemblies, gears, stem pinions, and differential side gears require loading fixtures for batch-quenched parts and quenching presses for ring gears. Because low unit cost must be maintained, quenching...
Series: ASM Handbook
Volume: 13C
Publisher: ASM International
Published: 01 January 2006
DOI: 10.31399/asm.hb.v13c.a0004209
EISBN: 978-1-62708-184-9
... restoration and tooth can lead to microorganism invasion, generation of acidic conditions, and the operation of differential aeration cells. Under these conditions, the bonding of the restoration to the dentinal walls through the underlying cement is likely to become weakened. In combination with biting...
Series: ASM Handbook
Volume: 20
Publisher: ASM International
Published: 01 January 1997
DOI: 10.31399/asm.hb.v20.a0002442
EISBN: 978-1-62708-194-8
... is texts by Haug, Kane, Greenwood, and Erdman ( Ref 1 , 2 , 3 , 4 , 5 , 6 ). The basic equations are not complex: a set of second-order differential equations (DEs) that define the force balance on each component of a system plus a set of constraint equations that tie the components together...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.a0005415
EISBN: 978-1-62708-196-2
... is the Lagrange multiplier. However, the derivatives of the corresponding grand function, for example, Eq 31 in a later section, are identically zero.) Because E is a functional, the derivatives are regarded as functional derivatives in the calculus of variations (any calculus of variations textbook may...
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0005189
EISBN: 978-1-62708-187-0
... energy is at a minimum, or equivalently from calculus: (Eq 4) d G ( P , T , n i … ) = 0 For systems in which chemical change can occur, dG can be expressed in partial differential form: (Eq 5) d G = ( ∂ G ∂ T ) P , n i … d T...
Series: ASM Handbook
Volume: 4F
Publisher: ASM International
Published: 01 February 2024
DOI: 10.31399/asm.hb.v04F.a0007006
EISBN: 978-1-62708-450-5
..., and differential side gears require loading fixtures for batch-quenched parts and quenching presses for ring gears. Because low unit cost must be maintained, quenching was integrated with heating, and handling facilities were arranged accordingly. Machined gears were charged into a continuous pusher-type gas...
Series: ASM Handbook Archive
Volume: 11
Publisher: ASM International
Published: 01 January 2002
DOI: 10.31399/asm.hb.v11.a0003530
EISBN: 978-1-62708-180-1
... on a weak plane. For the former, the extreme stresses can be found by taking Eq 11 and applying the calculus principle d σ X ′ / d θ = 0. That is, principal stresses are normal (perpendicular) stresses on planes for which the shear stresses are zero. The values of principal stresses are given by: (Eq...
Series: ASM Handbook
Volume: 11
Publisher: ASM International
Published: 15 January 2021
DOI: 10.31399/asm.hb.v11.a0006767
EISBN: 978-1-62708-295-2
... of a stress transformation is mainly to find the stresses that can be used in a failure criterion. These would be either the largest magnitude stresses in any direction or the magnitude of stresses on a weak plane. For the former, the extreme stresses can be found by taking Eq 11 and applying the calculus...
Series: ASM Handbook
Volume: 22A
Publisher: ASM International
Published: 01 December 2009
DOI: 10.31399/asm.hb.v22a.9781627081962
EISBN: 978-1-62708-196-2
Book: Casting
Series: ASM Handbook
Volume: 15
Publisher: ASM International
Published: 01 December 2008
DOI: 10.31399/asm.hb.v15.a0009020
EISBN: 978-1-62708-187-0
... that are in the ballpark of the final design. Casting engineers can learn this sense, and when they do, they become effective concurrent engineering partners in their customers' casting designs. Equation for Area Moment of Inertia <xref rid="a0009020-f1" ref-type="fig">(Fig. 1)</xref> Although the calculus...