Revision #3195 → #3708 · back to history
4028c4f4f80a| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Cartesian coordinates","snippet":"the Cartesian coordinates may be retrieved from the spherical coordinates"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}x&=r\\sin \\theta \\,\\cos \\varphi ,\\\\y&=r\\sin \\theta \\,\\sin \\varphi ,\\\\z&=r\\cos \\theta .\\end{aligned}}}"}] | [{"section":"Cartesian coordinates","snippet":"the Cartesian coordinates may be retrieved from the spherical coordinates"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}x&=r\\sin \\theta \\,\\cos \\varphi ,\\\\y&=r\\sin \\theta \\,\\sin \\varphi ,\\\\z&=r\\cos \\theta .\\end{aligned}}}"}] |
d98096938288| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The line element for an infinitesimal displacement from ( r , θ , φ ) to"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} \\mathbf {r} =\\mathrm {d} r\\,{\\hat {\\mathbf {r} }}+r\\,\\mathrm {d} \\theta \\,{\\hat {\\boldsymbol {\\theta }}}+r\\sin {\\theta }\\,\\mathrm {d} \\varphi \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} ,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\hat {\\mathbf {r} }}&=\\sin \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\sin \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}+\\cos \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\theta }}}&=\\cos \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}-\\sin \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\varphi }}}&=-\\sin \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\varphi \\,{\\hat {\\mathbf {y} }}\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle R={\\begin{pmatrix}\\sin \\theta \\cos \\varphi &\\sin \\theta \\sin \\varphi &{\\hphantom {-}}\\cos \\theta \\\\\\cos \\theta \\cos \\varphi &\\cos \\theta \\sin \\varphi &-\\sin \\theta \\\\-\\sin \\varphi &\\cos \\varphi &{\\hphantom {-}}0\\end{pmatrix}}.}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The line element for an infinitesimal displacement from ( r , θ , φ ) to"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} \\mathbf {r} =\\mathrm {d} r\\,{\\hat {\\mathbf {r} }}+r\\,\\mathrm {d} \\theta \\,{\\hat {\\boldsymbol {\\theta }}}+r\\sin {\\theta }\\,\\mathrm {d} \\varphi \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} ,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\hat {\\mathbf {r} }}&=\\sin \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\sin \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}+\\cos \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\theta }}}&=\\cos \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}-\\sin \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\varphi }}}&=-\\sin \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\varphi \\,{\\hat {\\mathbf {y} }}\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle R={\\begin{pmatrix}\\sin \\theta \\cos \\varphi &\\sin \\theta \\sin \\varphi &{\\hphantom {-}}\\cos \\theta \\\\\\cos \\theta \\cos \\varphi &\\cos \\theta \\sin \\varphi &-\\sin \\theta \\\\-\\sin \\varphi &\\cos \\varphi &{\\hphantom {-}}0\\end{pmatrix}}.}"}] |
668465486c88| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The linear transformation to this right-handed coordinate triplet is a rotation matrix"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} \\mathbf {r} =\\mathrm {d} r\\,{\\hat {\\mathbf {r} }}+r\\,\\mathrm {d} \\theta \\,{\\hat {\\boldsymbol {\\theta }}}+r\\sin {\\theta }\\,\\mathrm {d} \\varphi \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} ,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\hat {\\mathbf {r} }}&=\\sin \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\sin \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}+\\cos \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\theta }}}&=\\cos \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}-\\sin \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\varphi }}}&=-\\sin \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\varphi \\,{\\hat {\\mathbf {y} }}\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle R={\\begin{pmatrix}\\sin \\theta \\cos \\varphi &\\sin \\theta \\sin \\varphi &{\\hphantom {-}}\\cos \\theta \\\\\\cos \\theta \\cos \\varphi &\\cos \\theta \\sin \\varphi &-\\sin \\theta \\\\-\\sin \\varphi &\\cos \\varphi &{\\hphantom {-}}0\\end{pmatrix}}.}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The linear transformation to this right-handed coordinate triplet is a rotation matrix"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} \\mathbf {r} =\\mathrm {d} r\\,{\\hat {\\mathbf {r} }}+r\\,\\mathrm {d} \\theta \\,{\\hat {\\boldsymbol {\\theta }}}+r\\sin {\\theta }\\,\\mathrm {d} \\varphi \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} ,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\hat {\\mathbf {r} }}&=\\sin \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\sin \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}+\\cos \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\theta }}}&=\\cos \\theta \\cos \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\theta \\sin \\varphi \\,{\\hat {\\mathbf {y} }}-\\sin \\theta \\,{\\hat {\\mathbf {z} }},\\\\{\\hat {\\boldsymbol {\\varphi }}}&=-\\sin \\varphi \\,{\\hat {\\mathbf {x} }}+\\cos \\varphi \\,{\\hat {\\mathbf {y} }}\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle R={\\begin{pmatrix}\\sin \\theta \\cos \\varphi &\\sin \\theta \\sin \\varphi &{\\hphantom {-}}\\cos \\theta \\\\\\cos \\theta \\cos \\varphi &\\cos \\theta \\sin \\varphi &-\\sin \\theta \\\\-\\sin \\varphi &\\cos \\varphi &{\\hphantom {-}}0\\end{pmatrix}}.}"}] |
267edc9630d9| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The Cartesian unit vectors are thus related to the spherical unit vectors by"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}\\mathbf {\\hat {x}} \\\\\\mathbf {\\hat {y}} \\\\\\mathbf {\\hat {z}} \\end{bmatrix}}={\\begin{bmatrix}\\sin \\theta \\cos \\varphi &\\cos \\theta \\cos \\varphi &-\\sin \\varphi \\\\\\sin \\theta \\sin \\varphi &\\cos \\theta \\sin \\varphi &{\\hphantom {-}}\\cos \\varphi \\\\\\cos \\theta &-\\sin \\theta &{\\hphantom {-}}0\\end{bmatrix}}{\\begin{bmatrix}{\\boldsymbol {\\hat {r}}}\\\\{\\boldsymbol {\\hat {\\theta }}}\\\\{\\boldsymbol {\\hat {\\varphi }}}\\end{bmatrix}}}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The Cartesian unit vectors are thus related to the spherical unit vectors by"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{bmatrix}\\mathbf {\\hat {x}} \\\\\\mathbf {\\hat {y}} \\\\\\mathbf {\\hat {z}} \\end{bmatrix}}={\\begin{bmatrix}\\sin \\theta \\cos \\varphi &\\cos \\theta \\cos \\varphi &-\\sin \\varphi \\\\\\sin \\theta \\sin \\varphi &\\cos \\theta \\sin \\varphi &{\\hphantom {-}}\\cos \\varphi \\\\\\cos \\theta &-\\sin \\theta &{\\hphantom {-}}0\\end{bmatrix}}{\\begin{bmatrix}{\\boldsymbol {\\hat {r}}}\\\\{\\boldsymbol {\\hat {\\theta }}}\\\\{\\boldsymbol {\\hat {\\varphi }}}\\end{bmatrix}}}"}] |
d3252164119c| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The volume element spanning from r to r + d r , θ to θ + d θ , and φ to φ + d φ"},{"type":"math_alttext","value":"{\\displaystyle J={\\frac {\\partial (x,y,z)}{\\partial (r,\\theta ,\\varphi )}}={\\begin{pmatrix}\\sin \\theta \\cos \\varphi &r\\cos \\theta \\cos \\varphi &-r\\sin \\theta \\sin \\varphi \\\\\\sin \\theta \\sin \\varphi &r\\cos \\theta \\sin \\varphi &{\\hphantom {-}}r\\sin \\theta \\cos \\varphi \\\\\\cos \\theta &-r\\sin \\theta &{\\hphantom {-}}0\\end{pmatrix}},}"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} V=\\left|{\\frac {\\partial (x,y,z)}{\\partial (r,\\theta ,\\varphi )}}\\right|\\,\\mathrm {d} r\\,\\mathrm {d} \\theta \\,\\mathrm {d} \\varphi =r^{2}\\sin \\theta \\,\\mathrm {d} r\\,\\mathrm {d} \\theta \\,\\mathrm {d} \\varphi =r^{2}\\,\\mathrm {d} r\\,\\mathrm {d} \\Omega ~.}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The volume element spanning from r to r + d r , θ to θ + d θ , and φ to φ + d φ"},{"type":"math_alttext","value":"{\\displaystyle J={\\frac {\\partial (x,y,z)}{\\partial (r,\\theta ,\\varphi )}}={\\begin{pmatrix}\\sin \\theta \\cos \\varphi &r\\cos \\theta \\cos \\varphi &-r\\sin \\theta \\sin \\varphi \\\\\\sin \\theta \\sin \\varphi &r\\cos \\theta \\sin \\varphi &{\\hphantom {-}}r\\sin \\theta \\cos \\varphi \\\\\\cos \\theta &-r\\sin \\theta &{\\hphantom {-}}0\\end{pmatrix}},}"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {d} V=\\left|{\\frac {\\partial (x,y,z)}{\\partial (r,\\theta ,\\varphi )}}\\right|\\,\\mathrm {d} r\\,\\mathrm {d} \\theta \\,\\mathrm {d} \\varphi =r^{2}\\sin \\theta \\,\\mathrm {d} r\\,\\mathrm {d} \\theta \\,\\mathrm {d} \\varphi =r^{2}\\,\\mathrm {d} r\\,\\mathrm {d} \\Omega ~.}"}] |
b090ae21ba5c| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"leads to the following expressions for the gradient and Laplacian for scalar fields"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla f&={\\partial f \\over \\partial r}{\\hat {\\mathbf {r} }}+{1 \\over r}{\\partial f \\over \\partial \\theta }{\\hat {\\boldsymbol {\\theta }}}+{1 \\over r\\sin \\theta }{\\partial f \\over \\partial \\varphi }{\\hat {\\boldsymbol {\\varphi }}},\\\\[8pt]\\nabla ^{2}f&={1 \\over r^{2}}{\\partial \\over \\partial r}\\left(r^{2}{\\partial f \\over \\partial r}\\right)+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\partial f \\over \\partial \\theta }\\right)+{1 \\over r^{2}\\sin ^{2}\\theta }{\\partial ^{2}f \\over \\partial \\varphi ^{2}}\\\\[8pt]&=\\left({\\frac {\\partial ^{2}}{\\partial r^{2}}}+{\\frac {2}{r}}{\\frac {\\partial }{\\partial r}}\\right)f+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\frac {\\partial }{\\partial \\theta }}\\right)f+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\frac {\\partial ^{2}}{\\partial \\varphi ^{2}}}f~,\\\\[8pt]\\end{aligned}}}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"leads to the following expressions for the gradient and Laplacian for scalar fields"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla f&={\\partial f \\over \\partial r}{\\hat {\\mathbf {r} }}+{1 \\over r}{\\partial f \\over \\partial \\theta }{\\hat {\\boldsymbol {\\theta }}}+{1 \\over r\\sin \\theta }{\\partial f \\over \\partial \\varphi }{\\hat {\\boldsymbol {\\varphi }}},\\\\[8pt]\\nabla ^{2}f&={1 \\over r^{2}}{\\partial \\over \\partial r}\\left(r^{2}{\\partial f \\over \\partial r}\\right)+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\partial f \\over \\partial \\theta }\\right)+{1 \\over r^{2}\\sin ^{2}\\theta }{\\partial ^{2}f \\over \\partial \\varphi ^{2}}\\\\[8pt]&=\\left({\\frac {\\partial ^{2}}{\\partial r^{2}}}+{\\frac {2}{r}}{\\frac {\\partial }{\\partial r}}\\right)f+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\frac {\\partial }{\\partial \\theta }}\\right)f+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\frac {\\partial ^{2}}{\\partial \\varphi ^{2}}}f~,\\\\[8pt]\\end{aligned}}}"}] |
be0f6a393457| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"the following expressions for the divergence and curl of vector fields"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla f&={\\partial f \\over \\partial r}{\\hat {\\mathbf {r} }}+{1 \\over r}{\\partial f \\over \\partial \\theta }{\\hat {\\boldsymbol {\\theta }}}+{1 \\over r\\sin \\theta }{\\partial f \\over \\partial \\varphi }{\\hat {\\boldsymbol {\\varphi }}},\\\\[8pt]\\nabla ^{2}f&={1 \\over r^{2}}{\\partial \\over \\partial r}\\left(r^{2}{\\partial f \\over \\partial r}\\right)+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\partial f \\over \\partial \\theta }\\right)+{1 \\over r^{2}\\sin ^{2}\\theta }{\\partial ^{2}f \\over \\partial \\varphi ^{2}}\\\\[8pt]&=\\left({\\frac {\\partial ^{2}}{\\partial r^{2}}}+{\\frac {2}{r}}{\\frac {\\partial }{\\partial r}}\\right)f+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\frac {\\partial }{\\partial \\theta }}\\right)f+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\frac {\\partial ^{2}}{\\partial \\varphi ^{2}}}f~,\\\\[8pt]\\end{aligned}}}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"the following expressions for the divergence and curl of vector fields"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\nabla f&={\\partial f \\over \\partial r}{\\hat {\\mathbf {r} }}+{1 \\over r}{\\partial f \\over \\partial \\theta }{\\hat {\\boldsymbol {\\theta }}}+{1 \\over r\\sin \\theta }{\\partial f \\over \\partial \\varphi }{\\hat {\\boldsymbol {\\varphi }}},\\\\[8pt]\\nabla ^{2}f&={1 \\over r^{2}}{\\partial \\over \\partial r}\\left(r^{2}{\\partial f \\over \\partial r}\\right)+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\partial f \\over \\partial \\theta }\\right)+{1 \\over r^{2}\\sin ^{2}\\theta }{\\partial ^{2}f \\over \\partial \\varphi ^{2}}\\\\[8pt]&=\\left({\\frac {\\partial ^{2}}{\\partial r^{2}}}+{\\frac {2}{r}}{\\frac {\\partial }{\\partial r}}\\right)f+{1 \\over r^{2}\\sin \\theta }{\\partial \\over \\partial \\theta }\\left(\\sin \\theta {\\frac {\\partial }{\\partial \\theta }}\\right)f+{\\frac {1}{r^{2}\\sin ^{2}\\theta }}{\\frac {\\partial ^{2}}{\\partial \\varphi ^{2}}}f~,\\\\[8pt]\\end{aligned}}}"}] |
4b99048ce2ad| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"the inverse Jacobian in Cartesian coordinates is"},{"type":"math_alttext","value":"{\\displaystyle J^{-1}={\\begin{pmatrix}{\\dfrac {x}{r}}&{\\dfrac {y}{r}}&{\\dfrac {z}{r}}\\\\\\\\{\\dfrac {xz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {yz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {-\\left(x^{2}+y^{2}\\right)}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}\\\\\\\\{\\dfrac {-y}{x^{2}+y^{2}}}&{\\dfrac {x}{x^{2}+y^{2}}}&0\\end{pmatrix}}.}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"the inverse Jacobian in Cartesian coordinates is"},{"type":"math_alttext","value":"{\\displaystyle J^{-1}={\\begin{pmatrix}{\\dfrac {x}{r}}&{\\dfrac {y}{r}}&{\\dfrac {z}{r}}\\\\\\\\{\\dfrac {xz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {yz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {-\\left(x^{2}+y^{2}\\right)}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}\\\\\\\\{\\dfrac {-y}{x^{2}+y^{2}}}&{\\dfrac {x}{x^{2}+y^{2}}}&0\\end{pmatrix}}.}"}] |
40990114642f| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The metric tensor in the spherical coordinate system is"},{"type":"math_alttext","value":"{\\displaystyle J^{-1}={\\begin{pmatrix}{\\dfrac {x}{r}}&{\\dfrac {y}{r}}&{\\dfrac {z}{r}}\\\\\\\\{\\dfrac {xz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {yz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {-\\left(x^{2}+y^{2}\\right)}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}\\\\\\\\{\\dfrac {-y}{x^{2}+y^{2}}}&{\\dfrac {x}{x^{2}+y^{2}}}&0\\end{pmatrix}}.}"}] | [{"section":"Integration and differentiation in spherical coordinates","snippet":"The metric tensor in the spherical coordinate system is"},{"type":"math_alttext","value":"{\\displaystyle J^{-1}={\\begin{pmatrix}{\\dfrac {x}{r}}&{\\dfrac {y}{r}}&{\\dfrac {z}{r}}\\\\\\\\{\\dfrac {xz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {yz}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}&{\\dfrac {-\\left(x^{2}+y^{2}\\right)}{r^{2}{\\sqrt {x^{2}+y^{2}}}}}\\\\\\\\{\\dfrac {-y}{x^{2}+y^{2}}}&{\\dfrac {x}{x^{2}+y^{2}}}&0\\end{pmatrix}}.}"}] |
8a87eccdde96| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Distance and angle in spherical coordinates","snippet":"The distance between the two points can be expressed as"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\mathbf {r} }&=(r,\\theta ,\\varphi ),\\\\{\\mathbf {r} '}&=(r',\\theta ',\\varphi ')\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}D&={\\sqrt {r^{2}+r'^{2}-2rr'(\\sin {\\theta }\\sin {\\theta '}\\cos {(\\varphi -\\varphi ')}+\\cos {\\theta }\\cos {\\theta '})}}\\end{aligned}}}"}] | [{"section":"Distance and angle in spherical coordinates","snippet":"The distance between the two points can be expressed as"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}{\\mathbf {r} }&=(r,\\theta ,\\varphi ),\\\\{\\mathbf {r} '}&=(r',\\theta ',\\varphi ')\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}D&={\\sqrt {r^{2}+r'^{2}-2rr'(\\sin {\\theta }\\sin {\\theta '}\\cos {(\\varphi -\\varphi ')}+\\cos {\\theta }\\cos {\\theta '})}}\\end{aligned}}}"}] |
72a3a04c7323| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Kinematics","snippet":"the position of a point or particle"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =r\\mathbf {\\hat {r}} .}"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {v} ={\\frac {\\mathrm {d} \\mathbf {r} }{\\mathrm {d} t}}={\\dot {r}}\\mathbf {\\hat {r}} +r\\,{\\dot {\\theta }}\\,{\\hat {\\boldsymbol {\\theta }}}+r\\,{\\dot {\\varphi }}\\sin \\theta \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {a} ={}&{\\frac {\\mathrm {d} \\mathbf {v} }{\\mathrm {d} t}}\\\\[1ex]={}&{\\hphantom {+}}\\;\\left({\\ddot {r}}-r\\,{\\dot {\\theta }}^{2}-r\\,{\\dot {\\varphi }}^{2}\\sin ^{2}\\theta \\right)\\mathbf {\\hat {r}} \\\\&{}+\\left(r\\,{\\ddot {\\theta }}+2{\\dot {r}}\\,{\\dot {\\theta }}-r\\,{\\dot {\\varphi }}^{2}\\sin \\theta \\cos \\theta \\right){\\hat {\\boldsymbol {\\theta }}}\\\\&{}+\\left(r{\\ddot {\\varphi }}\\,\\sin \\theta +2{\\dot {r}}\\,{\\dot {\\varphi }}\\,\\sin \\theta +2r\\,{\\dot {\\theta }}\\,{\\dot {\\varphi }}\\,\\cos \\theta \\right){\\hat {\\boldsymbol {\\varphi }}}\\end{aligned}}}"}] | [{"section":"Kinematics","snippet":"the position of a point or particle"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =r\\mathbf {\\hat {r}} .}"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {v} ={\\frac {\\mathrm {d} \\mathbf {r} }{\\mathrm {d} t}}={\\dot {r}}\\mathbf {\\hat {r}} +r\\,{\\dot {\\theta }}\\,{\\hat {\\boldsymbol {\\theta }}}+r\\,{\\dot {\\varphi }}\\sin \\theta \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {a} ={}&{\\frac {\\mathrm {d} \\mathbf {v} }{\\mathrm {d} t}}\\\\[1ex]={}&{\\hphantom {+}}\\;\\left({\\ddot {r}}-r\\,{\\dot {\\theta }}^{2}-r\\,{\\dot {\\varphi }}^{2}\\sin ^{2}\\theta \\right)\\mathbf {\\hat {r}} \\\\&{}+\\left(r\\,{\\ddot {\\theta }}+2{\\dot {r}}\\,{\\dot {\\theta }}-r\\,{\\dot {\\varphi }}^{2}\\sin \\theta \\cos \\theta \\right){\\hat {\\boldsymbol {\\theta }}}\\\\&{}+\\left(r{\\ddot {\\varphi }}\\,\\sin \\theta +2{\\dot {r}}\\,{\\dot {\\varphi }}\\,\\sin \\theta +2r\\,{\\dot {\\theta }}\\,{\\dot {\\varphi }}\\,\\cos \\theta \\right){\\hat {\\boldsymbol {\\varphi }}}\\end{aligned}}}"}] |
30e32fc02f60| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Kinematics","snippet":"Its velocity is then"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =r\\mathbf {\\hat {r}} .}"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {v} ={\\frac {\\mathrm {d} \\mathbf {r} }{\\mathrm {d} t}}={\\dot {r}}\\mathbf {\\hat {r}} +r\\,{\\dot {\\theta }}\\,{\\hat {\\boldsymbol {\\theta }}}+r\\,{\\dot {\\varphi }}\\sin \\theta \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {a} ={}&{\\frac {\\mathrm {d} \\mathbf {v} }{\\mathrm {d} t}}\\\\[1ex]={}&{\\hphantom {+}}\\;\\left({\\ddot {r}}-r\\,{\\dot {\\theta }}^{2}-r\\,{\\dot {\\varphi }}^{2}\\sin ^{2}\\theta \\right)\\mathbf {\\hat {r}} \\\\&{}+\\left(r\\,{\\ddot {\\theta }}+2{\\dot {r}}\\,{\\dot {\\theta }}-r\\,{\\dot {\\varphi }}^{2}\\sin \\theta \\cos \\theta \\right){\\hat {\\boldsymbol {\\theta }}}\\\\&{}+\\left(r{\\ddot {\\varphi }}\\,\\sin \\theta +2{\\dot {r}}\\,{\\dot {\\varphi }}\\,\\sin \\theta +2r\\,{\\dot {\\theta }}\\,{\\dot {\\varphi }}\\,\\cos \\theta \\right){\\hat {\\boldsymbol {\\varphi }}}\\end{aligned}}}"}] | [{"section":"Kinematics","snippet":"Its velocity is then"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =r\\mathbf {\\hat {r}} .}"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {v} ={\\frac {\\mathrm {d} \\mathbf {r} }{\\mathrm {d} t}}={\\dot {r}}\\mathbf {\\hat {r}} +r\\,{\\dot {\\theta }}\\,{\\hat {\\boldsymbol {\\theta }}}+r\\,{\\dot {\\varphi }}\\sin \\theta \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {a} ={}&{\\frac {\\mathrm {d} \\mathbf {v} }{\\mathrm {d} t}}\\\\[1ex]={}&{\\hphantom {+}}\\;\\left({\\ddot {r}}-r\\,{\\dot {\\theta }}^{2}-r\\,{\\dot {\\varphi }}^{2}\\sin ^{2}\\theta \\right)\\mathbf {\\hat {r}} \\\\&{}+\\left(r\\,{\\ddot {\\theta }}+2{\\dot {r}}\\,{\\dot {\\theta }}-r\\,{\\dot {\\varphi }}^{2}\\sin \\theta \\cos \\theta \\right){\\hat {\\boldsymbol {\\theta }}}\\\\&{}+\\left(r{\\ddot {\\varphi }}\\,\\sin \\theta +2{\\dot {r}}\\,{\\dot {\\varphi }}\\,\\sin \\theta +2r\\,{\\dot {\\theta }}\\,{\\dot {\\varphi }}\\,\\cos \\theta \\right){\\hat {\\boldsymbol {\\varphi }}}\\end{aligned}}}"}] |
5d95f7067cbf| Field | From #3195 | To #3708 |
|---|---|---|
| anchors | [{"section":"Kinematics","snippet":"and its acceleration is"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =r\\mathbf {\\hat {r}} .}"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {v} ={\\frac {\\mathrm {d} \\mathbf {r} }{\\mathrm {d} t}}={\\dot {r}}\\mathbf {\\hat {r}} +r\\,{\\dot {\\theta }}\\,{\\hat {\\boldsymbol {\\theta }}}+r\\,{\\dot {\\varphi }}\\sin \\theta \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {a} ={}&{\\frac {\\mathrm {d} \\mathbf {v} }{\\mathrm {d} t}}\\\\[1ex]={}&{\\hphantom {+}}\\;\\left({\\ddot {r}}-r\\,{\\dot {\\theta }}^{2}-r\\,{\\dot {\\varphi }}^{2}\\sin ^{2}\\theta \\right)\\mathbf {\\hat {r}} \\\\&{}+\\left(r\\,{\\ddot {\\theta }}+2{\\dot {r}}\\,{\\dot {\\theta }}-r\\,{\\dot {\\varphi }}^{2}\\sin \\theta \\cos \\theta \\right){\\hat {\\boldsymbol {\\theta }}}\\\\&{}+\\left(r{\\ddot {\\varphi }}\\,\\sin \\theta +2{\\dot {r}}\\,{\\dot {\\varphi }}\\,\\sin \\theta +2r\\,{\\dot {\\theta }}\\,{\\dot {\\varphi }}\\,\\cos \\theta \\right){\\hat {\\boldsymbol {\\varphi }}}\\end{aligned}}}"}] | [{"section":"Kinematics","snippet":"and its acceleration is"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {r} =r\\mathbf {\\hat {r}} .}"},{"type":"math_alttext","value":"{\\displaystyle \\mathbf {v} ={\\frac {\\mathrm {d} \\mathbf {r} }{\\mathrm {d} t}}={\\dot {r}}\\mathbf {\\hat {r}} +r\\,{\\dot {\\theta }}\\,{\\hat {\\boldsymbol {\\theta }}}+r\\,{\\dot {\\varphi }}\\sin \\theta \\,\\mathbf {\\hat {\\boldsymbol {\\varphi }}} }"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\mathbf {a} ={}&{\\frac {\\mathrm {d} \\mathbf {v} }{\\mathrm {d} t}}\\\\[1ex]={}&{\\hphantom {+}}\\;\\left({\\ddot {r}}-r\\,{\\dot {\\theta }}^{2}-r\\,{\\dot {\\varphi }}^{2}\\sin ^{2}\\theta \\right)\\mathbf {\\hat {r}} \\\\&{}+\\left(r\\,{\\ddot {\\theta }}+2{\\dot {r}}\\,{\\dot {\\theta }}-r\\,{\\dot {\\varphi }}^{2}\\sin \\theta \\cos \\theta \\right){\\hat {\\boldsymbol {\\theta }}}\\\\&{}+\\left(r{\\ddot {\\varphi }}\\,\\sin \\theta +2{\\dot {r}}\\,{\\dot {\\varphi }}\\,\\sin \\theta +2r\\,{\\dot {\\theta }}\\,{\\dot {\\varphi }}\\,\\cos \\theta \\right){\\hat {\\boldsymbol {\\varphi }}}\\end{aligned}}}"}] |