Photo to Math

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You can paste or upload a photo of handwritten or printed math and convert it into editable expressions by hovering on it and hitting "Recognize."

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Example 1

Prop 1.4

① ((z_1)+(z_2))=(z_1)+(z_2)

② ((z_1)*(z_2))=(z_1)*(z_2)

③ z*z=z*z=|z|^2

④ |(z_1)*(z_2)|=|(z_1)|*|(z_2)|

⑤ 1/z=z/(z*z)=z/(|z|^2)=(x−i*y)/(x^2+y^2)=x/(x^2+y^2)−iy/(x^2+y^2)

Example 2

2.2. Iordanskii-Kogarko-Wijngaarden model

Consider an incompressible fluid of density (ρ_1)=const containing bubbles of compressible gas. We will consider the IKW system [31,35,55] respecting the following assumptions. First, all bubbles have the same radius R at a given point of space. Second, surface tension, viscosity and heat conduction are neglected. Third, the bubble radius is significantly smaller than the inter-bubble distance d which, in turn, is significantly smaller than the characteristic scale of the problem l:

R≪d≪l.

The governing equations are as follows:

∂(ρ)/∂(t)+div(ρ*u)=0,

(∂(ρ)*u)/∂(t)+div(ρ*u⊗u+p*I)=0,

p=(p_2)+(ρ_1)*(3/2*R^2+R*R),

∂(N)/∂(t)+div(N*u)=0.

Example 3

Example: For our velocity field, what are the streamline at t=0 ?

d(x)/d(s)=u(x,y,0)=1,

d(y)/d(s)=v(x,y,0)=−2*t*e^(−t^2)|=0.

∴y=const

x=s+(x_0).