Overlaps significantly with the previous two posts but here we go. Condensed version bc studying for midterm!!
The del operator is:
\vec{\nabla} = <\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}>
∇ ⃗ = < ∂ ∂ x , ∂ ∂ y , ∂ ∂ z > \vec{\nabla} = <\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}> ∇ =< ∂ x ∂ , ∂ y ∂ , ∂ z ∂ >
\vec{\nabla} \cdot \vec{v} ∇ ⃗ ⋅ v ⃗ \vec{\nabla} \cdot \vec{v} ∇ ⋅ v is the divergence of a vector field , and \vec{\nabla} \times \vec{v} ∇ ⃗ × v ⃗ \vec{\nabla} \times \vec{v} ∇ × v is the curl. The curl is: \vec{\nabla} \times \vec{v} = <\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}, \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}, \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}>
∇ ⃗ × v ⃗ = < ∂ v z ∂ y − ∂ v y ∂ z , ∂ v x ∂ z − ∂ v z ∂ x , ∂ v y ∂ x − ∂ v x ∂ y > \vec{\nabla} \times \vec{v} = <\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}, \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}, \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}> ∇ × v =< ∂ y ∂ v z − ∂ z ∂ v y , ∂ z ∂ v x − ∂ x ∂ v z , ∂ x ∂ v y − ∂ y ∂ v x > There's also a review of line, surface and volume integrals, and spherical and cylindrical coordinates. In lieu of typing all that up here's a screenshot of the inside cover page of the textbook, with lots of convenient equations:
Next there's two fundamental theorems.
Gauss' theorem says that the volume integral of divergence over a bounded space is equal to the surface integral of the field over its boundary:
\int_V \vec{\nabla} \cdot \vec{v} \: d\tau = \oint_S \vec{v} \cdot d \vec{a}
∫ V ∇ ⃗ ⋅ v ⃗ d τ = ∮ S v ⃗ ⋅ d a ⃗ \int_V \vec{\nabla} \cdot \vec{v} \: d\tau = \oint_S \vec{v} \cdot d \vec{a} ∫ V ∇ ⋅ v d τ = ∮ S v ⋅ d a Stokes' theorem says that the surface integral of curl over a bounded surface is equal to the line integral of the field over its boundary:
\int_S \vec{\nabla} \times \vec{v} \: d\vec{a} = \oint_P \vec{v} \cdot d \vec{l}
∫ S ∇ ⃗ × v ⃗ d a ⃗ = ∮ P v ⃗ ⋅ d l ⃗ \int_S \vec{\nabla} \times \vec{v} \: d\vec{a} = \oint_P \vec{v} \cdot d \vec{l} ∫ S ∇ × v d a = ∮ P v ⋅ d l Finally we have the dirac delta function , which is 0 everywhere except x=0, where it's infinity. More usefully, it's defined such that:
\int_{-\infty}^{\infty} \delta(x) dx = 1, \int_{-\infty}^{\infty} f(x) \delta(x) dx = f(0), \int_{-\infty}^{\infty} f(x) \delta(x-a) dx = f(a)
∫ − ∞ ∞ δ ( x ) d x = 1 , ∫ − ∞ ∞ f ( x ) δ ( x ) d x = f ( 0 ) , ∫ − ∞ ∞ f ( x ) δ ( x − a ) d x = f ( a ) \int_{-\infty}^{\infty} \delta(x) dx = 1, \int_{-\infty}^{\infty} f(x) \delta(x) dx = f(0), \int_{-\infty}^{\infty} f(x) \delta(x-a) dx = f(a) ∫ − ∞ ∞ δ ( x ) d x = 1 , ∫ − ∞ ∞ f ( x ) δ ( x ) d x = f ( 0 ) , ∫ − ∞ ∞ f ( x ) δ ( x − a ) d x = f ( a ) Eq. 1.93 says that, for expressions involving delta functions
D_1, D_2 D 1 , D 2 D_1, D_2 D 1 , D 2 , they are equal if \int_{-\infty}^{\infty} f(x) D_1(x) dx = \int_{-\infty}^{\infty} f(x) D_2(x) dx
∫ − ∞ ∞ f ( x ) D 1 ( x ) d x = ∫ − ∞ ∞ f ( x ) D 2 ( x ) d x \int_{-\infty}^{\infty} f(x) D_1(x) dx = \int_{-\infty}^{\infty} f(x) D_2(x) dx ∫ − ∞ ∞ f ( x ) D 1 ( x ) d x = ∫ − ∞ ∞ f ( x ) D 2 ( x ) d x for all
. There's also section 1.6 with some more theorems but I'll ignore that for now.
Samson Zhang