Question-220198

Question Number 220198 by SdC355 last updated on 07/May/25

Commented by SdC355 last updated on 08/May/25

i already know and understand about operator ▽^→ ×F^→ ▽^→ ×F^→ = determinant ((e_1 ^→ ,e_2 ^→ ,e_3 ^→ ),(((∂ )/∂x),((∂ )/∂y),((∂ )/∂z)),(F_x ,F_y ,F_z ))=((∂F_z /∂y)−(∂F_y /∂z))e_1 ^→ +((∂F_x /∂z)−(∂F_z /∂x))e_2 ^→ +((∂F_y /∂x)−(∂F_x /∂y))e_3 ^→ but i can′t understand process that... ′′ d(F_x dx+F_y dy+F_z dz)= (∂_y F_x −∂_z F_y )dy∧dz+(∂_z F_x −∂_x F_z )dz∧dx+(∂_x F_y −∂_y F_x )dx∧dy ′′ ∮_( C) F_x dx+F_y dy+F_z dz=∫∫_( D) (∂_y F_x −∂_z F_y )dy∧dx+(∂_z F_x −∂_x F_z )dz∧dx+(∂_x F_y −∂_y F_x )dx∧dy

$$\mathrm{i}\:\mathrm{already}\:\mathrm{know}\:\mathrm{and}\:\mathrm{understand}\:\mathrm{about}\:\:\mathrm{operator}\:\overset{\rightarrow} {\bigtriangledown}×\overset{\rightarrow} {\boldsymbol{\mathrm{F}}} \\ $$$$\overset{\rightarrow} {\bigtriangledown}×\overset{\rightarrow} {\boldsymbol{\mathrm{F}}}=\begin{vmatrix}{\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} }&{\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} }&{\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} }\\{\frac{\partial\:\:}{\partial{x}}}&{\frac{\partial\:\:}{\partial{y}}}&{\frac{\partial\:\:}{\partial{z}}}\\{{F}_{{x}} }&{{F}_{{y}} }&{{F}_{{z}} }\end{vmatrix}=\left(\frac{\partial{F}_{{z}} }{\partial{y}}−\frac{\partial{F}_{{y}} }{\partial{z}}\right)\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +\left(\frac{\partial\mathrm{F}_{{x}} }{\partial{z}}−\frac{\partial{F}_{{z}} }{\partial{x}}\right)\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +\left(\frac{\partial{F}_{{y}} }{\partial{x}}−\frac{\partial{F}_{{x}} }{\partial{y}}\right)\overset{\rightarrow} {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\mathrm{but}\:\mathrm{i}\:\mathrm{can}'\mathrm{t}\:\mathrm{understand}\:\mathrm{process}\:\mathrm{that}… \\ $$$$\:''\:\mathrm{d}\left({F}_{{x}} \mathrm{d}{x}+{F}_{{y}} \mathrm{d}{y}+{F}_{{z}} \mathrm{d}{z}\right)= \\ $$$$\left(\partial_{{y}} {F}_{{x}} −\partial_{{z}} {F}_{{y}} \right)\mathrm{d}{y}\wedge\mathrm{d}{z}+\left(\partial_{{z}} {F}_{{x}} −\partial_{{x}} {F}_{{z}} \right)\mathrm{d}{z}\wedge\mathrm{d}{x}+\left(\partial_{{x}} {F}_{{y}} −\partial_{{y}} {F}_{{x}} \right)\mathrm{d}{x}\wedge\mathrm{d}{y}\:'' \\ $$$$\oint_{\:{C}} \:{F}_{{x}} \mathrm{d}{x}+{F}_{{y}} \mathrm{d}{y}+{F}_{{z}} \mathrm{d}{z}=\int\int_{\:\mathcal{D}} \left(\partial_{{y}} {F}_{{x}} −\partial_{{z}} {F}_{{y}} \right)\mathrm{d}{y}\wedge\mathrm{d}{x}+\left(\partial_{{z}} {F}_{{x}} −\partial_{{x}} {F}_{{z}} \right)\mathrm{d}{z}\wedge\mathrm{d}{x}+\left(\partial_{{x}} {F}_{{y}} −\partial_{{y}} {F}_{{x}} \right)\mathrm{d}{x}\wedge\mathrm{d}{y} \\ $$

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