Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Bookshelves/Geometry/Modern_Geometry_(Bishop)The standard approach is to develop more results of advanced Euclidean geometry first and to eventually back up and go into hyperbolic geometry. In order to get where we need to get, we will not forma...The standard approach is to develop more results of advanced Euclidean geometry first and to eventually back up and go into hyperbolic geometry. In order to get where we need to get, we will not formally develop some of the advanced Euclidean results that are logically needed because we would never get to the "modern" part of the course’s name. We will come back and develop, or at least introduce conceptually, some of these when we need them late in the course.
- https://math.libretexts.org/Workbench/De_Anza-_Math_1D/02%3A_Multiple_Integration/2.05%3A_Cylindrical_and_Spherical_CoordinatesIn this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are u...In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.03%3A_Linear_TransformationsThis page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformation...This page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformations. It highlights the importance of the zero vector, standard coordinate vectors, and defines transformations like rotations, dilations, and the identity transformation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.3%3A_SimilarityThis page explores similar matrices defined by the relation A=CBC−1, focusing on their geometric interpretations, eigenvalues, eigenvectors, and properties as an equivalence relation. It expl...This page explores similar matrices defined by the relation A=CBC−1, focusing on their geometric interpretations, eigenvalues, eigenvectors, and properties as an equivalence relation. It explains how to compute matrix powers, emphasizing transformations and changes of coordinates between different systems. The relationship between matrices A and B is examined, highlighting how they share characteristics like trace and determinant but may differ in eigenvectors.
- https://math.libretexts.org/Workbench/De_Anza-_Math_1D/01%3A_Differentiation_of_Functions_of_Several_Variables/1.03%3A_Partial_Derivatives/1.3E%3A_Exercises_for_Section_1.3This page discusses exercises on calculating partial and higher-order derivatives of functions, including limit definitions, surface plot analysis, and practical applications like volume and area. It ...This page discusses exercises on calculating partial and higher-order derivatives of functions, including limit definitions, surface plot analysis, and practical applications like volume and area. It also covers problems in differential calculus, such as finding points where partial derivatives equal zero, verifying Laplace's and heat equations, and analyzing rates of change in contexts like dimensions and productivity functions. Answers to the various exercises are provided throughout.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/07%3A_Geometry_on_Surfaces/7.04%3A_The_Family_of_Geometries__(_X_k__G_k_)A strong connection exists between the family (Pk^2, Sk) of elliptic geometries with curvature k>0 and the family (Dk, Hk) of hyperbolic geometries with curvature k<0. The two families sport id...A strong connection exists between the family (Pk^2, Sk) of elliptic geometries with curvature k>0 and the family (Dk, Hk) of hyperbolic geometries with curvature k<0. The two families sport identical descriptions of the transformation group, identical descriptions of straight lines, identical arc-length and area formulas, as well as identical formulas for the area of a triangle.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.06%3A_Moments_and_Centers_of_MassIn this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have see...In this section, we consider centers of mass (also called centroids, under certain conditions) and moments. The basic idea of the center of mass is the notion of a balancing point. Many of us have seen performers who spin plates on the ends of sticks. The performers try to keep several of them spinning without allowing any of them to drop. Mathematically, that sweet spot is called the center of mass of the plate.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.00%3A_Prelude_to_Linear_Transformations_and_Matrix_AlgebraThis page explores the non-linear complexities of a robot arm's joint movements and hand positions through the transformation function f(θ,ϕ,ψ). It discusses the relationship between ma...This page explores the non-linear complexities of a robot arm's joint movements and hand positions through the transformation function f(θ,ϕ,ψ). It discusses the relationship between matrices and transformations, covering how transformations can be expressed with matrices, their properties, and how matrix multiplication relates to composition. The chapter culminates in understanding matrix arithmetic and solving matrix equations.
- https://math.libretexts.org/Workbench/De_Anza-_Math_1D/02%3A_Multiple_Integration/2.02%3A_Double_Integrals_over_General_Regions/2.2E%3A_ExercisesThis page covers the classification of regions in calculus as Type I and Type II for integral evaluations, detailing calculations for areas and volumes under specified functions. It includes practical...This page covers the classification of regions in calculus as Type I and Type II for integral evaluations, detailing calculations for areas and volumes under specified functions. It includes practical exercises on double integrals and explores geometric interpretations in three-dimensional space. Additionally, it discusses the lunes of Alhazen, proving their area is equivalent to that of a corresponding triangle.
- https://math.libretexts.org/Workbench/De_Anza-_Math_1D/03%3A_Vector_Calculus/3.04%3A_Greens_TheoremGreen’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region D in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region D in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve C and a double integral over the region enclosed by C.
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/04%3A_Geometry/4.01%3A_The_BasicsThe reader who has seen group theory will know that in addition to the three properties listed in our definition, the group operation must satisfy a property called associativity. In the context of tr...The reader who has seen group theory will know that in addition to the three properties listed in our definition, the group operation must satisfy a property called associativity. In the context of transformations, the group operation is composition of transformations, and this operation is always associative. So, in the present context of transformations, we omit associativity as a property that needs checking.
