Calculus III
Contents
3 Dimensional space
Partial derivatives
Multiple integrals
Vector Functions
Line integrals
Surface integrals
Vector operators
Applications
© The scientific sentence. 2010
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Calculus III:
Quadric Surfaces
Using Gnuplot
1. Ellipse
Implicit form:
x2/a2 + y2/b2 = 1
Parametric form:
x = a cos t
y = b sin t
t ∈ [- π, π]
Gnuplot:
reset
set grid
set parametric
unset border
set xtics axis
set ytics axis
set xzeroaxis
set yzeroaxis
plot [- pi: pi] 2*cos(t), sin(t)
2. Hyperbola
Implicit form:
x2/a2 - y2/b2 = 1
Parametric form:
x = ± a cosh t
y = b sinh t
t ∈ R
Gnuplot:
reset
set grid
set parametric
unset border
set xtics axis
set ytics axis
set xzeroaxis
set yzeroaxis
plot [-2:2] 2*cosh(t), sinh(t), -2*cosh(t), sinh(t)
3. Ellipsoid
Implicit form:
x2/a2 + y2/b2
+ z2/c2 = 1
Parametric form:
x = a cos u cos v
y = b sin u cos v
z = c sin v
u ∈ [- π, π], v ∈ [- π/2, π/2]
Gnuplot:
reset
set grid
set parametric
set isosamples 20
splot [-pi:pi][-pi/2:pi/2] cos(u)*cos(v), 2*sin(u)*cos(v), 3*sin(v)
4. Hyperboloid of one sheet
Implicit form:
x2/a2 + y2/b2
- z2/c2 = 1
Parametric form:
x = a cos u cosh v
y = b sin u cosh v
z = c sinh v
u ∈ [- π, π], v ∈ R
Gnuplot:
reset
set grid
set parametric
set isosamples 20
splot [-pi:pi][-2.5:2.5] cos(u)*cosh(v), sin(u)*cosh(v), sinh(v)
5. Hyperboloid of two sheets
Implicit form:
x2/a2 + y2/b2
- z2/c2 = - 1
Parametric form:
x = a cos u sinh v
y = b sin u sinh v
z = ±c cosh v
u ∈ [- π, π], v ∈ R+
Gnuplot:
reset
set grid
set parametric
set isosamples 30
set view equal
splot [-pi:pi][-2.5:2.5] cos(u)*sinh(v), sin(u)*sinh(v), cosh(v), cos(u)*sinh(v), sin(u)*sinh(v), -cosh(v)
6. Elliptic paraboloid
Implicit form:
z/c = x2/a2 + y2/b2
Gnuplot:
reset
set grid
set isosamples 30
splot [-2:2][-2:2] y**2 + x**2
7. Hyperbolic paraboloid
Implicit form:
z2/c2 = y2/b2 + x2/a2
Gnuplot:
reset
set grid
set isosamples 30
splot [-2:2][-2:2] y**2 - x**2
8. Elliptic cone
Implicit form:
z/c = y2/b2 - x2/a2
Parametric form:
x = v cos u
y = v sin u
z = v
u ∈ [- π, π], v ∈ R.
Gnuplot:
reset
set grid
set parametric
set isosamples 30
splot [-pi:pi][-5:5] v*cos(u), v*sin(u), v
9. Cylinder
Parametric form:
x = cos u
y = sinu
z = v
u , v ∈ R .
Gnuplot:
reset
set grid
set parametric
set ticslevel 0
set xtics 1
set ytics 1
set ztics 2
set hidden3d
set isosamples 30,30
set xzeroaxis lt 2 lw 2
set yzeroaxis lt 2 lw 2
set style line 3 lw 3
set xrange [-3:3]
set yrange [-3:3]
set zrange [-5:5]
set ylabel "Y"
set xlabel "X"
set zlabel "Z"
splot cos(u), sin(u),v
10. Torus
Parametric form:
x = (R + r cos u) cos v
y = (R + r cos u) sin v
z = r sin u
u , v ∈ [- π, π].
Gnuplot:
reset
set grid
set parametric
set isosamples 40
splot [-pi:pi][-pi:pi] (5 + cos(u))*cos(v), (5 + cos(u))*sin(v), sin(u)
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