DGM:010117Flat bump/no boundary jetDifferential Geometry Smooth Manifolds Flat Bump SmoothnessDifferential Geometry Smooth Manifolds Interior Jet ComputationPrivateLabels unavailablePrivate saving unavailableSmooth manifolds, atlases, and smooth mapsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010118Torus saddle det -98Differential Geometry Smooth Manifolds Periodic Smooth FunctionsDifferential Geometry Smooth Manifolds Hessian Saddle TestPrivateLabels unavailablePrivate saving unavailableSmooth manifolds, atlases, and smooth mapsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010219Exact exponential push/pullDifferential Geometry Smooth Manifolds Explicit PushforwardDifferential Geometry Smooth Manifolds Explicit PullbackPrivateLabels unavailablePrivate saving unavailableTangent and cotangent bundles and tensorsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010220Nonorthogonal metric musical inversesDifferential Geometry Smooth Manifolds Flat Sharp MapsDifferential Geometry Smooth Manifolds Metric Cauchy SchwarzPrivateLabels unavailablePrivate saving unavailableTangent and cotangent bundles and tensorsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010321Figure-eight immersion transverse crossingDifferential Geometry Smooth Manifolds Immersion Derivative TestDifferential Geometry Smooth Manifolds Transverse Double PointPrivateLabels unavailablePrivate saving unavailableImmersions, submersions, embeddings, and rank theoremsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010322Nonlinear submersion F=(x+y,exp(x+y)z) straightens globally to the (u,w) projection, with fiber (t,-t,2) through (1,-1,2)Differential Geometry Smooth Manifolds Submersion RankDifferential Geometry Smooth Manifolds Straightening CoordinatesPrivateLabels unavailablePrivate saving unavailableImmersions, submersions, embeddings, and rank theoremsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010323The helicoid H=(u cos v, u sin v, v) embeds with inverse (z, x cos z + y sin z) and at (2,pi/2) has metric diag(1,5)Differential Geometry Smooth Manifolds Parametrized ImmersionDifferential Geometry Smooth Manifolds First Fundamental FormPrivateLabels unavailablePrivate saving unavailableImmersions, submersions, embeddings, and rank theoremsDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010424The Jordan shear field X=(x+y)∂x+y∂y has complete flow e^t(x+ty,y), Jacobian determinant e^{2t}, and unique rest point 0Differential Geometry Smooth Manifolds Linear FlowDifferential Geometry Smooth Manifolds Liouville DeterminantPrivateLabels unavailablePrivate saving unavailableVector fields, flows, Lie brackets, and Lie derivativesDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010425The quadratic escape field X=x²∂x has flow a/(1-at), sign-dependent maximal intervals, and group law only on a common domainDifferential Geometry Smooth Manifolds Maximal Integral CurveDifferential Geometry Smooth Manifolds Finite Time BlowupPrivateLabels unavailablePrivate saving unavailableVector fields, flows, Lie brackets, and Lie derivativesDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010426The weighted plane volume e^x dx∧dy has Lie derivative 2ω along X=∂x+y∂y, matching the flow pullback factor e^{2t}Differential Geometry Smooth Manifolds Interior Product Lie DerivativeDifferential Geometry Smooth Manifolds Weighted DivergencePrivateLabels unavailablePrivate saving unavailableVector fields, flows, Lie brackets, and Lie derivativesDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010527The polynomial one-form (2xy+z)dx+x^2 dy+x dz is exact with potential x^2 y+xz and integrates to 5 from the origin to (1,2,3)View access optionsDifferential Geometry Smooth Manifolds Polynomial Exact One FormDifferential Geometry Smooth Manifolds Path Independence ExactPrivateLabels unavailablePrivate saving unavailableDifferential forms, orientations, integration, and Stokes theoremDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010528The decomposable two-form (dx1+2 dx3) wedge (dx2-dx4) pulls back along F(u,v)=(u+v,2u-v,u-2v,3u+v) to -9 du wedge dvView access optionsDifferential Geometry Smooth Manifolds Pullback Decomposable Two FormDifferential Geometry Smooth Manifolds Oriented Surface IntegralPrivateLabels unavailablePrivate saving unavailableDifferential forms, orientations, integration, and Stokes theoremDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010629On g=e^{2r^2}(dx^2+dy^2) the function x^2-y^2 has vanishing Laplacian, and the unit disk has area pi(e^2-1)/2View access optionsDifferential Geometry Smooth Manifolds Conformal GradientDifferential Geometry Smooth Manifolds Conformal Volume LengthPrivateLabels unavailablePrivate saving unavailableMetrics, connections, geodesics, and curvatureDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010630The paraboloid z=x^2+2y^2 has shape operator diag(2,4) at the origin under the upward Weingarten convention S=-dNView access optionsDifferential Geometry Smooth Manifolds Graph Shape OperatorDifferential Geometry Smooth Manifolds Principal Curvature GraphPrivateLabels unavailablePrivate saving unavailableMetrics, connections, geodesics, and curvatureDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded
DGM:010631On S^2(r) x R the mixed sectional curvatures vanish, and the plane spanned by (e1+e3)/sqrt(2) and e2 has curvature 1/(2r^2)View access optionsDifferential Geometry Smooth Manifolds Product Sectional CurvatureDifferential Geometry Smooth Manifolds Ricci Scalar ProductPrivateLabels unavailablePrivate saving unavailableMetrics, connections, geodesics, and curvatureDifferential Geometry / Smooth Manifolds · Mixed reviewProgress not loaded