Symbolic mathematical operations

In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

In writing constraints, the following symbols are used for mathematical operations ... 

The simplest form of a block diagram is the block and arrows diagram. It consists of a single block with one input and one output (Figure 5A). The block normally contains the name of the element (Figure 5B) or the symbol of a mathematical operation (Figure 5C) to be performed on the input to obtain the desired output. Arrows identify the direction of information or signal flow. 

A word of caution should be raised. The formidable process of conducting such spatial integration should be viewed as principally symbolic. Although the mathematical operations will be valid, it is rare that in our application they will be so complicated. Indeed, in most cases, these operations will be very simple. For example, iff(x,y,z, t) is uniform in space, then we simply have, for Equation (3.1),... 

Care the H in pH derives from the symbol for hydrogen, and is always given a big letter. The p is a mathematical operator, and is always small. 

It is often easy to measure the flux density, e.g., using a flowmeter, and then determine the hydraulic conductivity or diffusion coefficient by dividing the flux by the driving force. One of the most difficult problems is determining how to represent the driving force. The symbol V is called an operator, which signifies that some mathematical operation is to be performed upon whatever function follows. V means to take the gradient with respect to distance. For Darcy s law under saturated... 

Mathematical formulas are made up of symbols and operations. The operations need to be performed in the correct order so that you get the correct answer. 

Often we must compute values of quantities that are not simple functions of the space coordinates, such as the y component of the momentum, py, where Equation (E.4) is not applicable. To get around this, we say that corresponding to every classical variable, there is a quantum mechanical operator. An operator is a symbol that directs us to do some mathematical operation. For example, the momentum operators are... 

To obtain a complete drug concentration profile, both the input and disposition kinetics must be known or assumed. If the input is an intravenous bolus, zero or first order, and disposition is first order, then the input and disposition can be combined mathematically through the convolution operation, represented by the symbol. Mathematically, this is represented as... 

An operator is merely a symbol which indicates that a mathematical operation must be carried out upon the expression which follows it. Thus ... 

Mathematical operations have specific rules for the use of mathematical symbols with SI units. A space or a half-high dot represents the multiplication of units a negative exponent, horizontal line, or slash represents the division of units, and if these mathematical symbols appear in the same line, parentheses must differentiate them. The percent sign (%) denotes the number 0.01 or 1/100, so that 1%= 0.01, 30% = 0.30, and so forth. Arabic numerals with the appropriate SI or recognized unit indicate the values of quantities. Commas are not used to separate numbers into groups of three. If more than four digits appear on either side of the decimal point, a space Table 3. Prefixes. separates the groups of three. 

Numbers, unit symbols, and names have set rules for mixing and differentiation for clarity of text and mathematical operations. These include a space between a numerical value and its unit symbol, indicating clearly the number a symbol belongs to in a given mathematical calculation, and no mixing of unit symbols and names nor making calculations on unit names. Different symbols represent values and units and the unit symbol should follow the value symbol separated by a slash. SI requires the use of standardized mathematical symbols and the explicit writing of a quotient quantity. 

X and Y are symbols given to actual counting data with their respective counting uncertainties, crx and ay. The quantities would be expressed as X + mathematical operations on two numbers the relations are given below ... 

Convolution — The convolution (or faltung) of two functions, /(f) and g(t), of time is the mathematical operation f] f(t)g(t - r)dr or equivalently /0 f(f - r)g(r) dr. Sometimes a lower limit other than zero, such as -oo, is appropriate. When functions are folded together in this way they are said to have been convolved (not convoluted ), and the symbol /(f) g( t) may be used to indicate this. 

Leave a space before and after mathematical operators that function as verbs or conjunctions that is, they have numbers on both sides or a symbol for a variable on one side and a number on the other. 

When mathematical symbols are used as adjectives, that is, with one number that is not part of a mathematical operation, do not leave a space between the symbol and the number. 

Some standard usages and symbols for mathematical operations and constants need never be defined. They include the following ... 

Massieu function 48 mathematical constants 83, 90 mathematical functions 83 mathematical operators 84 mathematical symbols 81-86 matrices 83, 85 matrix element of operator 16 maxwell 115 Maxwell equations 123 mean free path 56 mean international ohm 114 mean international volt 114 mean ionic activity 58 mean ionic activity coefficient 58 mean ionic molality 58 mean life 22, 93 mean relative speed 56 mechanics classical 12 quantum 16 mega 74 melting 51 metre 70,71,110 micro 74 micron 110 mile 110 Miller indices 38 milli 74... 

An operator is an abbreviated expression representing a particular mathematical operation. Operators are used even in elementary mathematics although that particular term is rarely used in that connection thus fl is written for the successive multiplication of a by and again by a In the same way the symbols In and sin are operators. The calculation of the differential coefficient djdx is also a specific mathematical operation. Let this operation be represented by the operator D such that the action of the operator on a function e.g. is represented by Dx, the function being written on the right hand side of the operator viz... 

Mathematics, in the very broadest sense, is the systematic study of relationships in the physical world and relationships between symbols which need not pertain to the real world. In relation to the world, mathematics is the language of science. It operates within the laws and constraints of science as it examines physical phenomena. Unlike science, however, mathematics has no constraints. So in relation to symbols, mathematics can be considered a pure mental activity which is capable of generating new concepts within the mind unrelated to anything that presently exists. 

Usually, the problem is to find simultaneously V and the values o that satisfy the eigenvalue equation (2.1), the form of the operator having been previously established. An operator is a symbol telling us to carry out a certain mathematical operation on anything following it. An example is the differential operator O = d/dx. It is easily found that ip=x is not an eigenfunction of this operator... 

The simplest form of the block diagram is a single block, with one input and one output (see Figure 39). The interior of the block usually contains a description of or the name of the element or symbol for the mathematical operation to be performed on the input (/) to yield the output (O). The arrows represent the direction of flow of information or signal. Two such operations are presented in Figure 40. 

Symbols for physical quantities should be single-lettered using the Latin or Greek alphabet. The letters may be capital or lower case but should be printed in italic (slanted) type. Subscripts and superscripts may be added for clarity. All subscripts and superscripts are printed in Roman type (upright) except when these are symbols for physical quantities and therefore printed in italic type. Symbols for units should always be printed in Roman type. Similarly, symbols for chemical elements, elementary particles and mathematical operators (e.g. sin, exp, In, d/dr, etc.) are also printed in Roman type (see sections 1.3 and 1.6 in [1]). 

In addition to the four elementary arithmetic operations, there are some other important mathematical operations, many of which involve only one number. The magnitude, or absolute value, of a scalar quantity is a number that gives the size of the number irrespective of its sign. It is denoted by placing vertical bars before and after the symbol for the quantity. This operation means... 

Algebra is a branch of mathematics that was invented by Greek mathematicians and developed by Hindu, Arab, and European mathematicians. It was apparently the first branch of symbolic mathematics. Its great utility comes from the fact that letters are used to represent constants and variables and that operations are indicated by symbols such as x, /, and so on. Operations can be car-... 

In this chapter we have introduced symbolic mathematics, which involves the manipulation of symbols instead of performing numerical operations. We have presented the algebraic tools needed to manipulate expressions containing real scalar variables, real vector variables, and complex scalar variables. We have also introduced ordinary and hyperbolic trigonometric functions, exponentials, and logarithms. A brief introduction to the techniques of problem solving was included. 

Mathematica has a powerful capability to carry out symbolic mathematics on algebraic expressions and can solve equations symbolically. In addition to the arithmetic operations, the principal Mathematica statements for manipulating algebraic expressions are Expand ], Factor ], Simplify ], Together ], and Apart ]. The Expand statement multiplies factors and powers out to give an expanded form of the expression. The following input and output illustrate this action In l] =Clear a,x]... 

An operator is a symbol for carrying out a mathematical operation (see Chapter... 

A mathematical operator is a symbol standing for carrying out a mathematical operation or a set of operations. Operators are important in quantum mechanics, since each mechanical variable has a mathematical operator corresponding to it. Operator symbols can be manipulated symbolically in a way similar to the algebra of ordinary variables, but according to a different set of rules. An important difference between ordinary algebra and operator algebra is that multiplication of two operators is not necessarily commutative, so that if A and B are two operators, AB BA can occur. 

A mathematical operator is a symbol that stands for carrying out a mathematical operation on some function. For example, we can use the symbol d/dx or the symbol Dx to stand for the operation of differentiating with respect to x. We will usually assign a symbol to an operator that consists of a letter with a caret over it. When an operator operates on a function, the result will generally be... 

Although a mathematical operator is a symbol that stands for the carrying out of an operation, we can define an operator algebra in which we manipulate these symbols much as we manipulate variables and numbers in ordinary algebra. We define the sum of two operators by... 

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