Substitution Rule (u-substitution):
If u=g(x), then
∫f(g(x))g′(x)dx=∫f(u)du
7. Integration by Parts
Integration by Parts Formula:
∫udv=uv−∫vdu
where u and dv are chosen parts of the integrand.
This method is useful when the integrand is a product of two functions.
The choice of u and dv can significantly affect the complexity of the resulting integral.
Example:
Let u=x and dv=exdx.
Then, du=dx and v=ex.
Thus,
∫xexdx=xex−∫exdx=xex−ex+C
where C is the constant of integration.
8. Gradient, Divergence, Curl and Laplacian
Gradient
The gradient of a scalar function f(x,y,z) is a vector field that points in the direction of the greatest rate of increase of the function. It is denoted as:
∇f=(∂x∂f,∂y∂f,∂z∂f)
where ∇ is the del operator.
The gradient is used in optimization problems to find local maxima and minima of functions.
The function
f(x,y)=x2+y2
has gradient
∇f=(2x,2y)
which can be plotted as a vector field.
Divergence
The divergence of a vector field F=(Fx,Fy,Fz) is a scalar function that measures the rate at which "stuff" is expanding or contracting at a point. It is given by:
∇⋅F=∂x∂Fx+∂y∂Fy+∂z∂Fz
Divergence is used in fluid dynamics to describe the behavior of fluid flow.
The vector field
F(x,y)=(x2,−y2)
has divergence
∇⋅F=2x−2y
which can be plotted as a scalar field.
Curl
The curl of a vector field F=(Fx,Fy,Fz) is a vector that describes the rotation or "twisting" of the field. It is given by: