MATH2111 Higher Several Variable Calculus Week 1 …web.maths.unsw.edu.au/~potapov/2111_2015/week-1/math2111-lecture-1...MATH2111 Higher Several Variable Calculus Week 1 Part 1 Dr

MATH2111 Higher Several Variable Calculus

Week 1 Part 1

Dr Denis Potapov

School of Mathematics and StatisticsUniversity of New South Wales

Semester 1, 2016 [updated: March 14, 2016]

D Potapov (UNSW Maths & Stats) MATH2111 Semester 1, 2016 1 / 26

Curves and surfacesSummary of analytical approach to curves and surfaces

Several Variable Calculus studies Calculus Tools for the mapping

f : Rm 7→ Rn.

f : D 7→ Rn, D ⊆ Rm

Conventional geometrical interpretation of the mapping is as follows:

f : R 7→ R plane curve via graph of function;

f : R2 7→ R space surface via graph of function; or implicit plane curvevia f = const;

f : R3 7→ R implicit space surface via f = const.

f : R 7→ Rn parametric curve;

n = 2 plane curve;n = 3 space curve;

f : R2 7→ R3 parametric surface;

f : Rn 7→ Rn n = 2 plane vector �eld;n = 3 space vector �eld.


Plane curvesEllipse

Plot

f(t) = a cos t i+ b sin t j, t ∈ [−π, π].

If

x = a cos t and y = b sin t,

then (xa

)2+(yb

)2= 1.

The latter is the implicit equation ofellipse:

gnuplot> set parametric

gnuplot> plot [-pi:pi] cos(t), 2*sin(t)


Ellipse animation

ellipse-animation.pdf


Plane curveFunction graph

Plot

f(t) = (1+ t) i+4

3t2 j, t ∈ [−1, 3]

If

x = 1+ t and y =4

3t2

then

y =4

3(x − 1)2

The graph of this function is parabola.


Directory of known functions and curves

plane-curves-directory.pdf (to create)


Space curvesExample of helix

Plot

f(t) = cos t i+ sin t j+ t k,

t ∈ [−2π, 2π] .

gnuplot> set parametric

gnuplot> splot [-2*pi:2*pi] cos(u), sin(u), u


Helix animation

helix-animation.pdf


"ABC" curve

Plotf(t) = cos t i+ sin t j+ sin 3t k, t ∈ R

with GNUPLOT


Limits, the case of curvesSimple approach to limits in case of curves

Forf : I 7→ Rn, I ⊆ R

withf(t) = (x1(t), x2(t), . . . , xn(t)) , xi : I 7→ R.

De�nition

limt→a

f(t) =(limt→a

x1(t), . . . , limt→a

xn(t)).

Theorem (Algebraic properties of limits)

limt→a

[f(t) + g(t)

]= lim

t→af(t) + lim

t→ag(t)

limt→a

[λ(t) · f(t)

]=[limt→a

λ(t)]·[limt→a

f(t)], where λ : I 7→ R

limt→a

[f(t) · g(t)

]=[limt→a

f(t)]·[limt→a

g(t)]


Limits, the case of curves

limits-algebraic-theorem.canvas


Continuity in case of curves


De�nition

f is continuous at t = a, if and only if

limt→a

f(t) = f(a)

f is continuous on I if f is continuous for every t = a ∈ I

withf(t) = (x1(t), x2(t), . . . , xn(t)) , xi : I 7→ R

Theorem

f is continuous if and only if xi is continuous for every i = 1, 2, . . . , n


Continuity, proofs

continuity-per-component.canvas


Continuity in case of curves, II

Forf, g : I 7→ Rn and λ : I 7→ R

Theorem

If

f, g and λ

are continuous, then

f + g, λ · f and f · g

are continuous


Continuity, proofs

continuity-algebraic-theorem.canvas


Derivative in case of curves


De�nition

f is di�erentiable at t = a, if

f ′(a) := limt→a

f(t)− f(a)

t − a

exists;

f is di�erentiable on I if f is di�erentiable at every t = a ∈ I

withf(t) = (x1(t), x2(t), . . . , xn(t)) , xi : I 7→ R

Theorem

f is di�erentiable if and only if xi is di�erentiable for every i = 1, 2, . . . , n


Derivative, proofs

derivative-per-component.canvas


Derivative in case of curves, II

Forf, g : I 7→ Rn and λ : I 7→ R

Theorem

If

f, g and λ

are di�erentiable, then

f + g, λ · f and f · g

are di�erentiable


Derivative, proofs

derivative-algebraic-theorem.canvas


Tangent LineGeometry of derivative


the limit

f ′(a) = limt→a

f(t)− f(a)

t − a

if exists and non-zero, represents thedirection of the tangent line at f(a).The parametric equation of such line is

r(s) = f(a) + (s − a) · f ′(a), s ∈ R.

f(t) − f(a)

f(a)

f(t)


Tangent LineAnimation

tangent-animation.pdf


VelocityMechanical of derivative

For a position vector of a particle at time t ∈ I:

r : I 7→ Rn, I ⊆ R

the limit

v(t0) = limt→t0

r(t)− r(t0)

t − t0

if exists, represents the direction and magnitude of the instantaneous velocity ofthe particle at moment t = t0.Similarly, the limit

a(t0) = limt→a

v(t)− v(t0)

t − t0

if exists, represents the direction and magnitude of the instantaneous acceleration


Smooth and piecewise smooth curves


De�nition

f is smooth over I, if it is di�erentiable over I (except the end points) and

f ′(a) 6= 0, ∀a ∈ I.

f is piece-wise smooth over I, if there is a partition

I = I1 ∪ I2 ∪ . . . ∪ Ik

such that everyf : Ij 7→ Rn, j = 1, 2, . . . , k

is smooth


Smooth and piecewise smooth curvesCondition f′ 6= 0 important

Plot

f(t) = t3 i+ t2 k, t ∈ R.

Ifx = t3 and t = x

13 ,

theny = x

23 .

Note that,

f ′(t) = 3t2 i+ 2t j and f ′(0) = 0.

y = x2/3y = (−x)2/3


Smooth and piecewise smooth curvesCondition f′ 6= 0 important

Plotf(t) = t3 i+ t3 k, t ∈ R.

Ifx = t3 then y = x .

On the other hand

f ′(t) = 3t2 i+ 3t2 j and f ′(0) = 0.


Surfaces, visualisation

quadratic-surfaces-visualisation.pdf (to create)JKress-surface-visualisation.pdf


Documents

MATH2111 Higher Several Variable Calculus Week 1 …web.maths.unsw.edu.au/~potapov/2111_2015/week-1/math2111-lecture-1...MATH2111 Higher Several Variable Calculus Week 1 Part 1 Dr