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Formulário Oficial

Formulário Oficial da Cadeira (PDF)

Integrais

∫dx(x2+b)32=1bxx2+b\int{\cfrac{dx}{(x^2 +b)^{\frac{3}{2}}}} = \cfrac{1}{b}\cfrac{x}{\sqrt{x^2 +b}}
∫xdx(x2+b)32=−1x2+b\int{\cfrac{x dx}{(x^2 +b)^{\frac{3}{2}}}} = - \cfrac{1}{\sqrt{x^2 +b}}
∫xdxx2+b=x2+b\int{\cfrac{x dx}{\sqrt{x^2 + b}}} = \sqrt{x^2 +b}
∫dxx2+b=ln⁡(x+x2+b)\int{\cfrac{ dx}{\sqrt{x^2 + b}}} = \ln(x + \sqrt{x^2 +b})
∫dxx(x+a)=1aln⁡(xx+a)\int{\cfrac{ dx}{x (x + a)}} = \cfrac{1}{a} \ln(\cfrac{x}{x+a})

Coordenadas

Coordenadas Cartesianas (x,y,z)(x,y,z)

dl⃗=dx u⃗x+dy u⃗y+dz u⃗zd\vec l = dx \ \vec u_x + dy \ \vec u_y + dz \ \vec u_z

dS=dx dydS = dx \ dy

dv=dx dy dzdv = dx \ dy \ dz

∇⃗F=(∂F∂x,∂F∂y,∂F∂z)\vec \nabla F =( \cfrac{\partial F}{\partial x},\cfrac{\partial F}{\partial y},\cfrac{\partial F}{\partial z})

∇⃗⋅A⃗=∂Ax∂x+∂Ay∂y+∂Az∂z\vec \nabla \cdot \vec A = \cfrac{\partial A_x}{\partial x} + \cfrac{\partial A_y}{\partial y} + \cfrac{\partial A_z}{\partial z}

∇⃗×A⃗=(∂∂x,∂∂y,∂∂z)×(Ax,Ay,Az)\vec \nabla \times \vec A = (\cfrac{\partial }{\partial x}, \cfrac{\partial }{\partial y}, \cfrac{\partial }{\partial z}) \times (A_x, A_y, A_z)

Coordenadas Polares (r,θ)(r,\theta)

dl⃗=dr u⃗r+r dθ u⃗θd\vec l = dr \ \vec u_r + r \ d\theta \ \vec u_{\theta}

dS=r dr dθdS = r \ dr \ d\theta

Coordenadas Cilíndricas (r,θ,z)(r, \theta, z)

dl⃗=dr u⃗r+r dθ u⃗θ+dz u⃗zd \vec l = dr \ \vec u_r + r \ d \theta \ \vec u{\theta} + dz \ \vec u_z

dv=r dr dθ dzdv = r \ dr \ d \theta \ dz

∇⃗F=(∂F∂r,1r∂F∂θ,∂F∂z)\vec \nabla F = ( \cfrac{\partial F}{\partial r},\cfrac{1}{r}\cfrac{\partial F}{\partial \theta},\cfrac{\partial F}{\partial z})

∇⃗⋅A⃗=1r∂(r Ar)∂r+1r∂Aθ∂θ+∂Az∂z\vec \nabla \cdot \vec A = \cfrac{1}{r}\cfrac{\partial(r \ A_r)}{\partial r} + \cfrac{1}{r}\cfrac{\partial A_{\theta}}{\partial \theta} + \cfrac{\partial A_z}{\partial z}

∇⃗×A⃗=(1r∂Az∂θ−∂Aθ∂z)u⃗r+(∂Ar∂z−∂Az∂r)u⃗θ+(1r∂(r Aθ)∂r−1r∂Ar∂θ)u⃗z\vec \nabla \times \vec A = (\cfrac{1}{r} \cfrac{\partial A_z}{\partial \theta}- \cfrac{\partial A_\theta}{\partial z}) \vec u_r + (\cfrac{\partial A_r}{\partial z} - \cfrac{\partial A_z}{\partial r} ) \vec u_{\theta} + (\cfrac{1}{r} \cfrac{\partial(r \ A_{\theta})}{\partial r} - \cfrac{1}{r} \cfrac{\partial A_r}{\partial \theta} )\vec u_z

Coordenadas Esféricas (r,θ,ϕ)(r,\theta, \phi)

dl⃗=dr u⃗r+r dθ u⃗θ+r sinθ dϕ u⃗ϕd\vec l = dr \ \vec u_r + r \ d\theta \ \vec u_{\theta} + r \ sin \theta \ d \phi \ \vec u_{\phi}

dv=r2 dr sinθ dθ dϕdv = r^2 \ dr \ sin \theta \ d\theta \ d\phi

∇⃗F=(∂F∂r,1r∂F∂θ,1rsinθ∂F∂ϕ)\vec \nabla F = ( \cfrac{\partial F}{\partial r},\cfrac{1}{r}\cfrac{\partial F}{\partial \theta},\cfrac{1}{r sin \theta}\cfrac{\partial F}{\partial \phi})

∇⃗⋅A⃗=1r2∂∂r(r2 Ar)+1rsinθ∂∂θ(sinθAθ)+1rsinθ∂∂ϕ(Aϕ)\vec \nabla \cdot \vec A = \cfrac{1}{r^2} \cfrac{\partial}{\partial r} (r^2 \ A_r) + \cfrac {1}{r sin \theta} \cfrac{\partial}{\partial \theta} (sin \theta A_{\theta}) + \cfrac {1}{r sin \theta} \cfrac{\partial}{\partial \phi} (A_{\phi})

∇⃗×A⃗=(1rsinθ∂(sinθAϕ)∂θ−∂(sinθAθ)∂ϕ)u⃗r+1r(1sinθ∂Ar∂ϕ−∂(rAϕ)∂r)u⃗θ+1r(∂(rAθ)∂r−∂Ar∂θ)u⃗ϕ\vec \nabla \times \vec A = (\cfrac {1}{r sin \theta} \cfrac{\partial (sin \theta A_{\phi})}{\partial \theta} - \cfrac{\partial(sin \theta A_{\theta})} {\partial \phi} ) \vec u_r + \cfrac{1}{r} (\cfrac{1}{sin \theta} \cfrac{\partial A_r}{\partial \phi} - \cfrac{\partial(r A_{\phi})}{\partial r}) \vec u_{\theta} + \cfrac{1}{r} (\cfrac{\partial (r A_{\theta})}{\partial r} - \cfrac{\partial A_r}{\partial \theta}) \vec u_{\phi}

Teorema da Divergência

∫v∇⃗⋅A⃗ dv=∮SA⃗⋅n⃗ dS\int_v{\vec \nabla \cdot \vec A \ dv} = \oint_S{ \vec A \cdot \vec n \ dS}

Teorema de Stokes

∫S∇⃗×A⃗ dS⃗=∮ΓA⃗⋅dl⃗\int_S {\vec \nabla \times \vec A \ d\vec S} = \oint_{\Gamma} \vec A \cdot d \vec l

Identidades Vetoriais

∇⃗⋅(A⃗×B⃗)=B⃗⋅(∇⃗×A⃗)−A⃗⋅(∇⃗×B⃗)\vec \nabla \cdot (\vec A \times \vec B) = \vec B \cdot (\vec \nabla \times \vec A) - \vec A \cdot (\vec \nabla \times \vec B)

∇⃗⋅(∇⃗×A⃗)=0\vec \nabla \cdot (\vec \nabla \times \vec A) = 0

∇⃗×(∇⃗×A⃗)=∇⃗(∇⃗⋅A⃗)−∇2A⃗\vec \nabla \times (\vec \nabla \times \vec A) = \vec \nabla (\vec \nabla \cdot \vec A) - \nabla^2 \vec A

Eletrostática

E⃗=14πϵ0qr2 u⃗r\vec E= \cfrac{1}{4\pi \epsilon_0} \cfrac{q}{r^2} \ \vec u_r

14πϵ0=9×109N m2 C−2\cfrac{1}{4\pi \epsilon_0} = 9 \times 10^9 N \ m^2 \ C^{-2}

∮ΓE⃗⋅dl⃗=0\oint_{\Gamma} \vec E \cdot d\vec l = 0

∇×E⃗=0\nabla \times \vec E = 0

VP=∫PRefE⃗⋅dl⃗V_P = \int_P^{Ref} \vec E \cdot d \vec l

E⃗=−∇⃗V\vec E = - \vec \nabla V