Có 40+ tài liệu thuộc chủ đề "toán học trong kỹ thuật"
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whose values do not change under parallel translation and rotation of the coordinate axes, and the semi-invariant. whose value does not change under rotation of the coordinate axes.. The invariant Δ is called the large discriminant of equation (4.4.2.1). The invariant δ is called the small discriminant.. Table 1 presents the classification of second-order curves based on invariants.. Classification of...
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Point in rectangular Cartesian coordinate system.. of intersection of the perpendiculars with the axes by M X , M Y , and M Z , respectively. where OM X , OM Y , and OM Z are the signed lengths of the directed segments OM. Z of the axes OX, OY , and OZ, respectively, are called the coordinates of...
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Vectors a and b are collinear if and only if a × b = 0. The cross product of basis vectors is. If the vectors are given by their coordinates a = (a x , a y , a z ) and b = (b x , b y , b z. The area of the parallelogram spanned by vectors...
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Figure 4.40. Plane passing through a point and parallel to two straight lines.. Equation of plane passing through point and parallel to two straight lines.. The equation of the plane passing through a point M 1 (x 1 , y 1 , z 1 ) and parallel to two straight lines with direction vectors R 1 = (l 1 ,...
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If the straight lines are given by canonical equations, then the condition that they are parallel can be written as. Let us show that the lines x – 1. 4 are parallel to each other.. 2 = 2 4 , and hence the lines are parallel.. Conditions for two lines to be perpendicular.. Two straight lines given by vector parametric...
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The distance between them can be calculated by the formula. The condition that the determinant in the numerator in is zero is the condition for the two lines in space to meet.. The numerator of the fraction in is the volume of the parallelepiped spanned by the vectors r 1 – r 2 , R 1 , and R 2...
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a 11 x 2 + a 22 y 2 + a 33 z 2 + 2a 12 xy + 2a 13 xz + a 23 yz (4.7.2.9) corresponding to equation (4.7.2.1) and its characteristic equation. The invariants S, T , and δ can be expressed in terms of the roots λ 1 , λ 2 , and λ 3 as...
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b n of this expansion, one first divides f (x) by x – c with remainder. Then one divides g 0 (x) by x – c with remainder. Expand the polynomial f(x. x 4 – 5x 3 – 3x 2 + 9 in powers of the difference x – 3 (c = 3).. We write out Horner’s scheme, where the...
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A constant K is called an upper bound for the real roots of equation (5.1.5.1) or the polynomial P n (x) if equation (5.1.5.1) has no real roots greater than or equal to K. in a similar way, one defines a lower and an upper bound for positive and negative roots of an equation or the corresponding polynomial.. K 1...
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If a matrix is real (i.e., all its entries are real), then the corresponding transpose and the adjoint matrix coincide.. A square matrix A is said to be normal if A ∗ A = AA. A normal matrix A is said to be unitary if A ∗ A = AA. The trace of a square matrix A ≡ [a ij...
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The integer r satisfying these two conditions is called the rank of the matrix A and is denoted by r = rank (A). Any nonzero rth-order minor of the matrix A is called its basic minor. The rows and the columns whose intersection yields its basic minor are called basic rows and basic columns of the matrix. The rank of...
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0 is called the characteristic equation of the matrix A of size n × n, and f A (λ) is called its characteristic polynomial. The spectrum of the matrix A (i.e., the set of all its eigenvalues) coincides with the set of all roots of its characteristic equation. The multiplicity of every root λ i of the characteristic equation is...
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Let x be any element of the space V with the coordinates (x 1. Real Euclidean Space. Definition and properties of a real Euclidean space.. A real Euclidean space (or simply, Euclidean space) is a real linear space V endowed with a scalar product (also called inner product), which is a real-valued function of two arguments x V , y...
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General square system of linear equations.. A square system of linear equations has the form. If the determinant of system (5.5.2.2) is different from zero, i.e. det A ≠ 0, then the system has a unique solution,. If the determinant of the matrix of system (5.5.2.2) is different from zero, i.e. Δ = det A ≠ 0, then the system...
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The rank of a linear operator A is the dimension of its range: rank (A. Properties of the rank of a linear operator:. Let A : V → V be a linear operator. Let A L(V , V) be a bounded linear operator in a Hilbert space V . The operator A ∗ in L(V , V) is called its...
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The number m is called the order of the associated vector x.. n l = n) in V consisting of eigenvectors and associated vectors of the operator A such that the action of the operator A is determined by the relations. l) are eigenvectors of the operator A corresponding to the eigenvalues λ k. The matrix A of the linear...
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A point x 0 on a smooth surface S is called a stationary point of a differentiable function f defined on S if the derivative of f at the point x 0 in any direction on S is equal to zero. The value f (x 0 ) of the function f at a stationary point x 0 is called its...
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Such a mapping is called an isomorphism or isomorphic mapping of the group G 1 onto the group G 2. An isomorphic mapping of a group G onto itself is called an automorphism of G. If f 1 : G → G and f 2 : G → G are two automorphisms of a group G, one can define another...
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L., Hornsby, J., and Schneider, D. C., The Theory of Matrices, Phoenix Edition, Dover Publications, New York, 2004.. and Pilz, G., The Concise Handbook of Algebra, Kluwer Academic, Dordrecht, Boston, 2002.. Edition, Dover Publications, New York, 1991.. R., Group Theory, Dover Publications, New York, 1987.. Edition, Dover Publications, New York, 1977.. Edition, Dover Publications, New York, 1999.. There is a...
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Asymptotes of the graph of a function.. An asymptote of the graph of a function y = f (x) is a straight line whose distance from the point (x, y) on the graph of y = f (x) tends to zero if at least one of the coordinates (x, y) tends to zero.. The line x = a is a...