http://matrix.skku.ac.kr/Cal-Book/ http://sage.skku.edu/
http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html
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Define Function Taylor expansion
Equation solving Approximate solution Partial fraction
Limit Right hand limit Left hand limit Limit at + infinity Limit at - infinity
Derivative Second derivative Indefinite integral Definite integral
Series
Radius of convergence
Define a vector Norm of the vector Distance from yz-plane Distance from zx-plane Distance from xz-plane Distance from x-axis Distance from y-axis Distance from z-axis
Define a vector valued function Velocity Acceleration Speed Velocity vector at t = 2 Acceleration vector at t = 2 Arc length Dot product Cross product |
f(x)=exp(-2*x) f(x).taylor(x, 2, 4)
solve(f(x)==0, x) find_root(f(x), a, b) f.partial_fraction(x)
limit(f(x), x=2) limit(f(x), x=2, dir='+') limit(f(x), x=2, dir='-') limit(f(x), x=+oo) limit(f(x), x=-oo)
diff(f(x), x) diff(f(x), x, 2) integral(f(x), x) integral(f(x), (x, -2, 3))
sum((e/10)^x, x, 1, +oo)
u(x)=1/(x*2^x) rho=limit(abs(u(x+1)/u(x)), x=+oo) 1/rho
x = vector([3, -4, 5]) x.norm() sqrt(x[0]^2) sqrt(x[1]^2) sqrt(x[2]^2) sqrt(x[1]^2 + x[2]^2) sqrt(x[0]^2 + x[2]^2) sqrt(x[0]^2 + x[1]^2)
var('t'); r=vector([2-5*t,4*t,1+3*t]) v=diff(r, t) a=diff(r, t, 2) v.norm() v.subs(t=2) a.subs(t=2) integral(v.norm(), (t, 0, 5)) v.dot_product(a) v.cross_product(a) |
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Find gradient
Double integral Triple integral
Double integral (in Polar coordinate)
Cylindrical to Rectangular
Spherical to Rectangular
Graph in plane Parametric function in plane Implicit function in plane Line segment in plane
Graph in Parametric function in Implicit function in
Level curve
Vector field Vector field in
Line integral
curl (A) div (A)
Green’s theorem Stokes’ theorem Divergence theorem |
var('x, y, z') f=x*y+y*z^2+x*z^3 f.gradient() integral(integral(f, (x, 0, y)), (y, 0, 1)) integral(integral(integral(f, (x, 0, y)), (y, 0, 1)), (z, 0, 1))
var('r, t') f=arctan(tan(t)) integral(integral(f*r, (r, 1, 2)), (t, 0, pi/4))
T = Cylindrical('height', ['radius', 'azimuth']) T.transform(radius=1, azimuth= pi, height=2)
T = Spherical('radius', ['azimuth', 'inclination']) T.transform(radius=3, azimuth=pi/6, inclination=pi/6)
plot(sin(x), (x, -4, 4)) parametric_plot((sin(x), cos(x)), (x, 0, 2*pi)) var('x, y'); implicit_plot(sin(x)-y==1, (x, -2, 2), (y, -2, 2)) line([(1, 1), (2, 2)], color='red')
var('x, y'); plot(x^2+y^2, (x, -2, 2), (y, -2, 2)) parametric_plot3d((sin(x), cos(x), x), (x, 0, 2*pi)) var('x, y, z'); implicit_plot3d(x^2+y^2==5, (x, -3, 3), (y, -3, 3), (z, -3, 3))
var('x, y'); contour_plot(x^2+y^2, (x, -1, 1), (y, -1, 1), cmap='hsv', labels=True)
var('x, y'); plot_vector_field((x+y, x), (x, -3, 3), (y, -3, 3))
var('x, y, z'); plot_vector_field3d((0, 0, 1), (x, -3, 3), (y, -3, 3), (z, -3, 3))
var('t'); r=vector([t-t^3, t^2, 0]) integral(r.dot_product(diff(r, t)), (t, 0, pi))
A(x,y,z) = P*i+Q*j+R*k # conservative if curl(F)=0. curlA=(diff(R,y)-diff(Q,z))*i+(diff(P,z)-diff(R,x))*j+(diff(Q,x)-diff(P,y))*k divA = diff(P,x)+diff(Q,y)+diff(R,z) # divergence of A
integral(integral((diff(N, x)-diff(M, y))*r, (r, 0, 3)), (t, 0, 2*pi)) integral(integral(curl(F).dot_product(-n), (r, 0, 1)), (t, 0, 2*pi)) integral(integral(integral(Div, 0, 3), (y, 0, 2)), (z, 0, 1)) |
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Rules for Inequalities
1. If 2. If 3. If 4. If 5. If 6. If 7. If 8. If 9. If
Special Functions
1. Exponential Functions If The number http://matrix.skku.ac.kr/Mobile-Sage-G/sage-grapher.html
2. Logarithmic Functions The logarithmic with base
3. Inverse Trigonometric Functions
4. Hyperbolic Functions
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5. Inverse Hyperbolic Functions
Formulas of Trigonometric Functions
1. Addition and Subtraction Formulas
2. Double Angle Formulas
3. Half Angle Formulas
4. Triple Angle Formulas
5. Product-to-Sum
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6. Sun-to-Product
7. Linear Combination
Limits http://matrix.skku.ac.kr/cal-lab/SKKU-Cell-Epsilon-Delta.html http://matrix.skku.ac.kr/cal-lab/cal-Limit.html
1. Limit Laws Suppose that the limits
(If http://matrix.skku.ac.kr/cal-lab/cal-2-1-7.html http://matrix.skku.ac.kr/cal-lab/cal-2-2-3.html
2. Squeeze Theorem (or Sandwich Theorem) If
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Derivatives 1. Differentiation Rules
2. Chain Rule If
3. Parametric Formula For the parametric equations:
4. Implicit Function Let
5. Inverse Function
6. Trigonometric Functions |
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7. Inverse Trigonometric Functions http://matrix.skku.ac.kr/cal-lab/cal-3-2-13.html
8. Logarithmic Functions
9. Exponential Functions
10. Hyperbolic Functions
11. Inverse Hyperbolic Functions
General Rules of Integration http://matrix.skku.ac.kr/cal-lab/cal-RiemannSum.html |
1. Basic Forms
http://matrix.skku.ac.kr/cal-lab/cal-7-7-Exm-8.html
http://matrix.skku.ac.kr/cal-lab/cal-5-2-20.html
2. Trigonometric Forms http://matrix.skku.ac.kr/cal-lab/cal-5-4-exm-7.html http://matrix.skku.ac.kr/cal-lab/cal-8-3-3.html
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3. Hyperbolic Forms
http://matrix.skku.ac.kr/cal-lab/cal-8-1-9.html
4. Forms Involving
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5. Inverse Trigonometric Forms
http://myhandbook.info/form_integ.html
Series
1. Taylor Series
http://matrix.skku.ac.kr/cal-lab/cal-10-5-Exm-11.html
2. Maclaurin Series
3. Binomial Series If
where
Vectors
1. Dot Product Let
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, then 
.
, then 
.
, then 
.
, then 
.
, then 
.
, then 
.
, then 
.
, then 
.
, then 
.
and 
, then a function of the form 
is called an exponential 
is called the base and 
is called the exponential.
and 
, is defined by 
.
and 
exist and 
is a constant. Then
if 
, where 
is a positive integer.
, where 
is a positive integer. 
is even, we require 
)
when 
is near 
and
then 
where 
is a positive integer and 
is a constant.
is any real number and 
is differentiable and 
, then 
.
and 
,
, 
be an implicit function. 
where 
.
or 
, 
,
,
, 
, 
, 
and 
are any real numbers and 
is a positive integer, we have 
and 
and 
.
onto 
: 
,
onto 
: 
and 
are two sides of a parallelogram, 
.
, 
where 
is a scalar
, then 
,
, 
: a vector equation
: parametric equations
: a symmetric equation
: a standard form of a plane
: a vector version of a plane
: parametric equations of a plane
be a potential function for 
. 
from 
and 
,
has a piecewise smooth boundary 
with positive orientation,
is 
div 
.