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| Spivak M. — Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus |
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| Предметный указатель |
 26
Absolute differential form 126
Absolute tensor 126
Absolute value 1
Algebra, fundamental theorem of 105
Alternating tensor 78
Analytic function 105
angle 4
Angle, preserving 4
Angle, solid 131
Approximation 15
Archimedes 137
Area 56
Area, element of 126
Area, surface 126 127
Basis, usual for  3
Bilinear function 3 23
Boundary of a chain 97 98
Boundary of a manifold-with-boundary 113
Boundary, of a set 7
Buoyant force 137
Cauchy integral formula 106
Cauchy integral theorem 106
Cauchy — Riemann equations 105
Cavalieri’s principle 62
Chain 97 100
Chain rule 19 32
Change of variable 67—72
Characteristic function 55
Closed curve 106
Closed differential form 92
Closed rectangle 5
Closed set 5
Compact 7
complex numbers 104
Complex variables 105
Component function 11 87
composition 11
Cone, generalized 131
Consistent choices of orientation 117
Constant function 20
Constraints 122
Content 56
Content zero 51
Continuous differential form 88
Continuous function 12
Continuous vector field 87
Continuously differentiable 31
Convergence 137
Coordinate condition 111
Coordinate system 111
Coordinate system, polar 73
Counterclockwise orientation 134
Cover 7
Cross product 84
Cube, singular 97
Cube, standard n-cube 97
Curl 88 137
Curve 97
Curve, closed 106
Curve, differentiable 96
Degenerate singular cube 105
Derivative 16
Derivative, directional 33
Derivative, partial 25
Derivative, partial, higher-order (mixed) 26
Derivative, partial, second-order (mixed) 26
Diffeomorphism 109
Differentiable = 88
Differentiable curve 96
Differentiable differential form 88
Differentiable differential form on a manifold 117
Differentiable function 15 16 105
Differentiable function, continuously 31
Differentiable vector field 87
Differentiable vector field on a manifold 115
Differential 91
Differential form 88
Differential form, absolute 126
Differential form, closed 92
Differential form, continuous 88
Differential form, differentiable 88
Differential form, exact 92
Differential form, on a manifold 117
Differential form, on a manifold, differentiable 117
Dimension of a manifold 109
Dimension of a manifold-with-boundary 113
Directional derivative 33
Distance 4
Divergence of a field 88 137
Divergence theorem 135
Domain 11
Dual space 5
Element of area 126
Element of length 126
Element of volume see "Volume element"
End point 87
Equal up to nth order 18
Euclidean space 1
Exact differential form 92
Exterior of a set 7
Faces of a singular cube 98
Field see "Vector field"
FORM see "Differential form"
Fubini’s Theorem 58
Function 11
Function, 26
Function, analytic 105
Function, characteristic 55
Function, component 11 87
Function, composition of 11
Function, constant 20
Function, continuous 12
Function, continuously differentiable 31
Function, differentiable 15 16 105
Function, homogeneous 34
Function, identity 11
Function, implicitly defined 41 see
Function, integrable 48
Function, inverse 11 34—39 see
Function, projection 11
Fundamental theorem of algebra 105
Fundamental theorem of calculus 100—104
Gauss 134
Generalized cone 131
grad f 96
Graph 11 115
Green’s Theorem 134
Half-space 113
Heine — Borel theorem 7
Homogeneous function 34
Homotopy 108
Identity function 11
Implicit function theorem 41
Implicitly defined function 41
Incompressible fluid 137
Independence of parameterization 104
Induced orientation 119
| Inequality see "Triangle inequality"
Inner product 2 77
Inner product, preserving 4
Inner product, usual 77 87
Integrable function 48
integral 48
Integral Formula, Cauchy 106
Integral of a form on a manifold 123—124
Integral of a form over a chain 101
Integral over a set 55
Integral over an open set 65
Integral theorem, Cauchy 106
Integral, iterated 59 60
Integral, line 101
Integral, lower 58
Integral, surface 102
Integral, upper 58
Interior of a set 7
Inverse function 11 34—39
Inverse function theorem 35
Irrotational fluid 137
Iterated integral 59 60
Jacobian matrix 17
Jordan-measurable 56
Kelvin 74
Lac locus 106
Lagrange’s method 122
Lagrangian multiplier 122
Leibnitz’s rule 62
Length 56 126
Length = norm 1
Length, element of 126
LIMIT 11
Line 1
Line integral 101
Linking number 132
Liouville 74
Lower integral 58
Lower sum 47
Manifold 109
Manifold-with-boundary 113
Manifold-with-corners 131 137
Mathematician (old style) 74
Matrix 1
Matrix, Jacobian 17
Matrix, transpose of 23 83
Maxima 26—27
Measure zero 50
Minima 26—27
Moebius strip 119 120 130
Multilinear function 23 75
Multiplier see "Lagrangian multiplier"
Norm 1
Norm preserving 4
normal see "Outward unit normal"
Notation 3 44 89
One-one (1-1) function 11
One-sided surface 121
Open cover 7
Open rectangle 5
Open set 5
Orientable manifold 119
Orientation 82 119
Orientation, consistent choices of 117
Orientation, counterclockwise 134
Orientation, induced 119
Orientation, usual 83 87 121
Orientation-preserving 118 123
Oriented manifold 119
Orthogonal vectors 5
Orthonormal basis 77
Oscillation 13
Outward unit normal 119 120
Parameterization, independence of 104
Partial derivative 25
Partial derivative, higher-order (mixed) 26
Partial derivative, second-order (mixed) 26
Partition of a closed interval 46
Partition of a closed rectangle 46
Partition of unity 63
Perpendicular 5
Plane 1
Poincare lemma 94
Point 1
Polar coordinate system 73
Polarization identity 5
Positive definiteness 3 77
PRODUCT see "Cross product" "Inner "Tensor "Wedge
Projection function 11
Rectangle (closed or open) 5
Refine a partition 47
Rotation of F 137
Sard’s theorem 72
Self-adjoint 85
Sign of a permutation 78
Singular n-cube 97
Solid angle 131
Space 1 see "Euclidean "Half-space" "Tangent
Sphere 111
Standard n-cube 97
Star-shaped 93
Stokes’ Theorem 102 124 135
Subordinate 63
Subrectangles of a partition 46
surface 127
Surface area 126 127
Surface integral 102
symmetric 2 77
Tangent space 86 115
Tangent vector 96
Tensor 75
Tensor product 75
Tensor, absolute 126
Tensor, alternating 78
Torus 115
Transpose of a matrix 23 83
Triangle inequality 4
Unit outward normal 119 120
Upper integral 58
Upper sum 47
Usual see "Basis" "Inner "Orientation"
Variable, change of 67—72
Variable, complex see "Complex variables"
Variable, function of n 11
Variable, independent of the first 18
Variable, independent of the second 17
Vector 1
Vector field 87
Vector field on a manifold 115
Vector field on a manifold, continuous 87
Vector field on a manifold, differentiable 115
Vector field, continuous 87
Vector field, differentiable 87
Vector, tangent 96
Vector-valued function 11
Volume 47 56 126
Volume element 83 126
Wedge product 79
Winding number 104
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