Revision #1131 → #3977 · back to history
617258f2981d| Field | From #1131 | To #3977 |
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| mathlib.module | Mathlib.Geometry.Manifold.VectorBundle.Tangent | Mathlib.Geometry.Manifold.IsManifold.Basic |
0686d00cd41c| Field | From #1131 | To #3977 |
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| anchors | [{"section":"Covariant and contravariant components of a vector with a metric","snippet":"there is a unique reciprocal basis"},{"type":"math_alttext","value":"{\\displaystyle g(Y^{i},X_{j})=\\delta _{j}^{i},}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}v&=\\sum _{i}v^{i}[\\mathbf {f} ]X_{i}=\\mathbf {f} \\,\\mathbf {v} [\\mathbf {f} ]\\\\&=\\sum _{i}v_{i}[\\mathbf {f^{\\sharp }} ]Y^{i}=\\mathbf {f} ^{\\sharp }\\mathbf {v} ^{\\sharp }[\\mathbf {f} ].\\end{aligned}}}"}] | [{"section":"Covariant and contravariant components of a vector with a metric","snippet":"there is a unique reciprocal basis"},{"type":"math_alttext","value":"{\\displaystyle g(Y^{i},X_{j})=\\delta _{j}^{i},}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}v&=\\sum _{i}v^{i}[\\mathbf {f} ]X_{i}=\\mathbf {f} \\,\\mathbf {v} [\\mathbf {v} ]\\\\&=\\sum _{i}v_{i}[\\mathbf {f^{\\sharp }} ]Y^{i}=\\mathbf {f} ^{\\sharp }\\mathbf {v} ^{\\sharp }[\\mathbf {f} ].\\end{aligned}}}"}] |
064c98951509| Field | From #1131 | To #3977 |
|---|---|---|
| anchors | [{"section":"Corollaries","snippet":"(2). Proof: let's suppose"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}&\\mathbf {e} ^{i}\\cdot \\mathbf {e} _{k}=(A^{ij}\\mathbf {e} _{j})\\cdot \\mathbf {e} _{k}\\\\\\xrightarrow {\\text{(D.3,D.5)}} {}&A^{ij}g_{jk}=\\delta _{k}^{i};\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}&g^{mk}A^{ij}g_{jk}=g^{mk}\\delta _{k}^{i}\\\\\\xrightarrow {\\text{(D.5)}} {}&\\;g^{mk}g_{kj}A^{ij}=g^{mi}\\\\\\xrightarrow {\\text{(1)}} {}&\\;\\delta _{j}^{m}A^{ij}=g^{mi}\\\\\\longrightarrow {}&\\;A^{im}=g^{mi}{\\stackrel {\\text{(D.6)}}{=}}g^{im}.\\quad \\blacksquare \\end{aligned}}}"}] | [{"section":"Corollaries","snippet":"(2). Proof: let's suppose"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}&\\mathbf {e} ^{i}\\cdot \\mathbf {e} _{k}=(A^{ij}\\mathbf {e} _{j})\\cdot \\mathbf {e} _{k}\\\\\\xrightarrow {\\text{(D.3,D.5)}} {}&A^{ij}g_{jk}=\\delta _{k}^{i};\\end{aligned}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}&g^{mk}A^{ij}g_{jk}=g^{mk}\\delta _{k}^{i}\\\\\\xrightarrow {\\text{(D.5)}} {}&\\;g^{mk}g_{kj}A^{ij}=g^{mi}\\\\\\xrightarrow {\\text{(1)}} {}&\\;\\delta _{j}^{m}A^{ij}=g^{mi}\\\\\\longrightarrow {}&\\;A^{im}=g^{mi}{\\stackrel {\\text{(D.6)}}{=}}g^{im}.\\quad \\blacksquare \\end{aligned}}}"}] |
e11c6600049c| Field | From #1131 | To #3977 |
|---|---|---|
| anchors | [{"section":"Corollaries","snippet":"(7). Proof:"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {v} \\cdot \\mathbf {e} _{i}{\\stackrel {\\text{(D.4)}}{{}={}}}(v_{j}\\mathbf {e} ^{j})\\cdot \\mathbf {e} _{i}{\\stackrel {\\text{(D.3)}}{{}={}}}v_{j}\\delta _{i}^{j}=v_{i}.\\quad \\blacksquare \\end{aligned}}}"}] | [{"section":"Corollaries","snippet":"(7). Proof:"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {v} \\cdot \\mathbf {e} _{i}{\\stackrel {\\text{(D.4)}}{{}={}}}(v_{j}\\mathbf {e} ^{j})\\cdot \\mathbf {e} _{i}{\\stackrel {\\text{(D.3)}}{{}={}}}v_{j}\\delta _{i}^{j}=v_{i}.\\quad \\blacksquare \\end{aligned}}"}] |
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