Function gradient
is given by:
Cartesian:
Spherical:
Tangent line of
at a is:
1st order Linear ODE
Parametric surfaces:
Elliptical paraboloid:
Elliptical cylinder:
-To find tangent vector, find parameters and sub into derivative.-To find normal vector, take cross product of partial derivatives with subbed parameters.
Tangent plane is given by:
Solving IVP’s:
Standard model:
Law of cooling:
Resistors:
Capacitors:
Inductors:
Double integrals:
-Can be permutated:
Type 1 region:
Type 2 region:
Arc length is given by:
Surface area is:
Independence:
-Functions are linearly independent if they cannot be expressed as a combination of each other.-A set of functions are linearly independent if their Wronskian is 0.Homogenous
solutions:
-Distinct root:
-Multiple root:
-Complex root:
To change function variable, multiply by
Jacobian:
Cartesian function on
polar rectangle:
Conversion from
polar:
Conversion to
polar:
Jacobian =
Conversion to
spherical:
Jacobian = For , equilibrium points, are constant solutions.
If and are continuous at , then , has a unique solution.An n th order IVP is equivalent to the 1 st
order IVP:
Triple integral
Jacobian is given by:
A direction field is the
plot of the vectors:
If k=0:
Approximation methods:
Euler method:
Absolute error:
Relative error:
Improved Euler method:
‘
Runge-Kutta method:
-A function can be determined from a vector field by testing various points.
- defines a gradient vector field.
Ratio test is:
If L>1: divergent.
If L<1: convergent.
If L=1: inconclusive.
Solve L<1 for x to find radius of convergence.-Summation index of a power series can be shifted, enabling addition of series’.For a change of variables to t, bounds are 0 ≤ t ≤ 1.For a function over a smooth curve, the line
integral is:
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