Math : Partial / Gradeint/ Directional / Gradient op/ Laplace op
Partial derivative
https://www.youtube.com/watch?v=kdMep5GUOBw
https://www.youtube.com/watch?v=dfvnCHqzK54
finding partial derivative = finding a slope
partial derivative with respect to x is like below
and the partial derivative with respect to y is like below
the partial derivative with respect to x
The symmetry of second partial derivatives
f(x,y)
Δf = ∇·∇f
formula for Laplacian
The formal definition of partial derivatives
Gradient
∇ = is being a vector full of partial derivative operators
Gradient and graphs
Gradient ∇f is shown like this on the graph.
* which means the direction of the steepest ascent.
Directional derivative
Divergence formula
only x component gradient value.
only y component gradient value
Laplacian intuition
Δf(x,y)
div(grad f) = ∇ㆍ∇f
what does the divergence represent?
 |
| gradient / direction to increase the slope |
case of divergence positive and negetive
and in and out is 0 divergence
* Δf(x,y) is like how much of a minimum point is this X, Y.
Laplacian computation example
f(x,y) = 3 + cos(x/2) sin(y/2)
divergence of the gradient is laplacian
Explicit Laplacian formula
f (x₁, x₂ , --- , xn) Δf = ∇·∇f
Curl 1 | Partial derivatives, gradient, divergence, curl | Multivariable Calculus
∇ × V = div(V)
∇ × V = curl(V) which means the measure of how much is that field rotating.
Curl example
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