Math cheat sheet

The Essence of Multivariable Calculus | #SoME3
The Essence of Multivariable Calculus | #SoME3

Math cheat sheet

University: Nanyang Technological University

Course: Mathematics 2 (MH1811)

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Students shared 88 documents in this course

MH1811 Math 2 Cheat Sheet

PART 1: MULTIVARIABLE CALCULUS

Level curve: ÿ(ý,þ)=�㕘, level surface: ÿ(ý,þ,ÿ)=�㕘

Curves and Surfaces with Their Equations:

Line:

ÿý+Āþ=ā

Circle:

(ý2ÿ)2+(þ2Ā)2=ÿ2

Ellipse:

ÿ(ý2ÿ)2+Ā(þ2Ā)2=ÿ2

Plane:

ÿý+Āþ+āÿ =Ă or (�㔫2�㔫�㗎)∙�㔧=0

Sphere:

(ý2ÿ)2+(þ2Ā)2+(ÿ2ā)2=ÿ2

Ellipsoid:

ÿ(ý2ÿ)2+Ā(þ2Ā)2+ā(ÿ2ā)2=ÿ2

1) Limits, Path Limits, Continuity

Continuity: lim

(ý,þ)→(ÿ,Ā)ÿ(ý,þ)=ÿ(ÿ,Ā)

Some Methods to Evaluate Limit:

(1) Substitution for continuous functions.

(2) Using Path Limits. A limit does not exist if limit

along path ÿ does not exist, or if limits along

different paths ÿ1 and ÿ2 have different values.

(3) Squeeze Theorem. If ÿ(ý,þ)fĀ(ý,þ)fℎ(ý,þ)

and lim

(ý,þ)→(ÿ,Ā)ÿ(ý,þ)= lim

(ý,þ)→(ÿ,Ā)ℎ(ý,þ)=�㔿

then lim

(ý,þ)→(ÿ,Ā)Ā(ý,þ)=�㔿.

2) Partial Derivatives, Gradient Vector

Definition of Partial Derivatives:

ÿý=�㔕ÿ

�㔕ý =lim

�㕡→ýÿ(ā,þ)2ÿ(ý,þ)

ā2ý

(ÿý is also rate of change along positive-ý direction)

ÿþ=�㔕ÿ

�㔕þ=lim

�㕡→þÿ(ý,ā)2ÿ(ý,þ)

ā2þ

(ÿþ is also rate of change along positive-þ direction)

Alternative way: ÿý=Ă

Ăý(ÿ(ý,þ)), ÿþ=Ă

Ăþ(ÿ(ý,þ))

(treating þ or ý as a constant, respectively)

Gradient Vector: ∇ÿ(ý,þ)=(ÿý,ÿþ)=�㔕ÿ

�㔕ýÿ+�㔕ÿ

�㔕þĀ

Second Partial Derivatives:

ÿýý =�㔕

�㔕ý(�㔕ÿ

�㔕ý)=�㔕2ÿ

�㔕ý2, ÿýþ =�㔕

�㔕þ(�㔕ÿ

�㔕ý)= �㔕2ÿ

�㔕þ�㔕ý

ÿþý =�㔕

�㔕ý(�㔕ÿ

�㔕þ)= �㔕2ÿ

�㔕ý�㔕þ, ÿþþ =�㔕

�㔕þ(�㔕ÿ

�㔕þ)=�㔕2ÿ

�㔕þ2

(Clairaut’s Theorem: ÿýþ =ÿþý if continuous)

Hessian Matrix: �㔻(ÿ)=[ÿýý ÿýþ

ÿþý ÿþþ]

Chain Rule: (Case 1: ý(ā), þ(ā), Case 2: ý(Ā,ā), þ(Ā,ā))

Case 1: Ăÿ

Ăā =�㔕ÿ

�㔕ýĂý

Ăā +�㔕ÿ

�㔕þĂþ

Ăā

Case 2: �㔕ÿ

�㔕Ā=�㔕ÿ

�㔕ý�㔕ý

�㔕Ā+�㔕ÿ

�㔕þ�㔕þ

�㔕Ā, �㔕ÿ

�㔕ā =�㔕ÿ

�㔕ý�㔕ý

�㔕ā+�㔕ÿ

�㔕þ�㔕þ

�㔕ā

Implicit Differentiation: (�㔹(ý,þ,ÿ)=ÿ)

�㔕ÿ

�㔕ý=2�㔹ý

�㔹ÿ, �㔕ÿ

�㔕þ=2�㔹þ

�㔹ÿ

Directional Derivatives: (unit vector �㔮=(Ă1,Ă2))

Ā�㔮ÿ(ÿ,Ā)= lim

ℎ→0ÿ(ÿ+ℎĂ1,Ā+ℎĂ2)2ÿ(ÿ,Ā)

=∇ÿ(ÿ,Ā)∙�㔮

(a) Maximum rate of change =‖∇ÿ(ÿ,Ā)‖

and it occurs in the direction �㔮= ∇ÿ(ÿ,Ā)

‖∇ÿ(ÿ,Ā)‖

(b) Minimum rate of change =2‖∇ÿ(ÿ,Ā)‖

and it occurs in the direction �㔮=2 ∇ÿ(ÿ,Ā)

‖∇ÿ(ÿ,Ā)‖

(Note: ‖∇ÿ(ÿ,Ā)‖=√ÿý2+ÿþ2)

Applications of Gradient Vector:

(1) Tangent Plane to Level Surface (�㔹(ý,þ,ÿ)=�㕘)

∇�㔹(ý0,þ0,ÿ0)∙(ý2ý0,þ2þ0,ÿ2ÿ0)=0

(2) Normal Line to Level Surface (�㔹(ý,þ,ÿ)=�㕘)

�㔫=(ý0,þ0,ÿ0)+ā∇�㔹(ý0,þ0,ÿ0)

(3) Tangent Plane to Graph ÿ=ÿ(ý,þ)

ÿ=ÿ(ý0,þ0)+ÿý(ý0,þ0)(ý2ý0)+ÿþ(ý0,þ0)(þ2þ0)

(4) Normal Line to Graph ÿ=ÿ(ý,þ) → change the

equation into �㔹(ý,þ,ÿ)=�㕘, then use (2).

(5) Linear Approximation for ÿ(ý,þ) at (ÿ,Ā)

ÿ(ý,þ)jÿ(ÿ,Ā)+ÿý(ÿ,Ā)(ý2ÿ)+ÿþ(ÿ,Ā)(þ2Ā)

(6) Total Differential (estimate error, sensitivity, etc.)

∆ÿjĂÿ =ÿý(ÿ,Ā)∆ý+ÿþ(ÿ,Ā)∆þ

3) Optimization

Types of critical points:

(a) Stationary: ∇ÿ=�㗎 (i.e. ÿý=0 and ÿþ=0)

(b) Singular: either ÿý or ÿþ (or both) does not exist

Saddle: stationary but not local extremum (max/min)

Some Methods of Optimization:

(1) Second Derivative Test (for interior points).

1. Find all stationary points (∇ÿ(ÿ,Ā)=�㗎).

2. Evaluate Ā=|ÿýý ÿýþ

ÿþý ÿþþ|=ÿýýÿþþ 2(ÿýþ)2

a. Ā>0, ÿýý >0 → ÿ(ÿ,Ā) is a local minimum.

b. Ā>0, ÿýý <0 → ÿ(ÿ,Ā) is a local maximum.

c. Ā<0 → saddle point (ÿ(ÿ,Ā) is not max/min).

(2) Lagrange Multiplier (for boundary points).

1. Show that Global Extreme values exist, i.e. if

ÿ(ý,þ) is continuous and region is closed and

bounded, then ÿ has global max and min.

2. Solve { ∇ÿ(ý,þ)=�㔆∇Ā(ý,þ)

Ā(ý,þ)=�㕘

3. From all points, find global max and min points.

(3) Lagrangian Function (for boundary points).

1. Show that Global Extreme values exist.

2. Lagrangian function:

�㔿(ý,þ,…,�㔆)=ÿ(ý,þ,…)2�㔆Ā(ý,þ,…)

3. Solve �㔿ý=0, �㔿þ=0, &, 2�㔿�㔆=0.

4. From all points, find global max and min points.

4) Double Integrals

(1) Rectangles: �㕅={(ý,þ) | ÿfýfĀ,āfþfĂ}

∬ÿ(ý,þ) Ă�㔴

�㕅=∫ ∫ ÿ(ý,þ) ĂþĂý

Ă

ā

Ā

ÿ=∫ ∫ ÿ(ý,þ) ĂýĂþ

Ā

ÿ

Ă

ā

(2) Type I: �㕅={(ý,þ) | ÿfý fĀ, Ā1(ý)fþfĀ2(ý)}

∬ÿ(ý,þ) Ă�㔴

�㕅=∫ ∫ ÿ(ý,þ) ĂþĂý

Ā2(ý)

Ā1(ý)

Ā

ÿ

(3) Type II: �㕅={(ý,þ) | āfþfĂ, ℎ1(þ)fýfℎ2(þ)}

∬ÿ(ý,þ) Ă�㔴

�㕅=∫ ∫ ÿ(ý,þ) ĂýĂþ

ℎ2(þ)

ℎ1(þ)

Ă

ā

To change order of integration, sketch the region of

integration and express it as the other type of region.

PART 2: SEQUENCES AND SERIES

1) Sequences

Convergence: lim

Ā→∞ÿĀ is a finite value.

Otherwise, the sequence diverges.

Some Methods to Evaluate Convergence:

(1) Limit Techniques: rationalizing, dividing by highest

power, squeeze theorem, etc. can be applied.

A theorem: If lim

Ā→∞|ÿĀ|=0, then lim

Ā→∞ÿĀ=0

(2) Using Subsequences. lim

Ā→∞ÿĀ does not exist if

there is a divergence subsequence, or if there are

convergent subsequences with different limits.

(3) Fitting with a Function (ÿĀ=ÿ(Ā))

If lim

ý→∞ÿ(ý)=�㔿 or ±∞, then lim

Ā→∞ÿĀ=�㔿 or ±∞.

(By using functions, we can use L’Hospital’s Rule.)

(4) Monotone Convergence Theorem. Every bounded

monotonic sequence is convergent.

2) Series

Partial Sum: Ā�㕁=ÿ1+ÿ2+⋯+ÿĀ=∑ÿ�㕁

�㕁

Ā=1

Convergence: or lim

�㕁→∞Ā�㕁 is a finite value.

Otherwise, the series diverges.

Some Types of Series:

(1) Geometric Series (ÿÿĀ)

a.

|ÿ|<1 → convergent, Ā=ÿ/(12ÿ)

b.

|ÿ|g1 → divergent

(2) Telescoping Series (ÿĀ2ÿĀ+ÿ)

If lim

Ā→∞ÿĀ=�㔿, Ā=(∑ÿĀ

ÿ

Ā=1 )2ÿ�㔿

(3) p-Series (1/Ā�㕝)

a. �㕝>1 → convergent

b. �㕝f1 → divergent, e.g. harmonic series (1/Ā)

(4) Alternating Series ((21)ĀĀĀ or (21)Ā21ĀĀ)

To test convergence, use Alternating Series Test.

a.

∑|ÿĀ| converges → absolutely convergent

b.

∑|ÿĀ| div, ∑ÿĀ conv → conditionally convergent

c. ∑|ÿĀ|, ∑ÿĀ diverges → divergent

Some Methods to Evaluate Convergence:

(1) Test for Divergence

If lim

Ā→∞ÿĀb0 or does not exist, ∑ÿĀ diverges.

(2) Alternating Series Test

If lim

Ā→∞ĀĀ=0 and ĀĀ+1 fĀĀ, ∑ÿĀ converges.

(3) Integral Test (ÿĀ=ÿ(Ā))

1. Prove ÿ(ý) is continuous, positive, decreasing.

2. Evaluate integral ∫ÿ(ý) Ăý

�㕁.

a. Integral converges → ∑ÿĀ converges

b. Integral diverges → ∑ÿĀ diverges

(4) The Comparison Test

a. ÿĀfĀĀ and ∑ĀĀ converges → ∑ÿĀ converges

b. ÿĀgĀĀ and ∑ĀĀ diverges → ∑ÿĀ diverges

(5) Limit Comparison Test

a. lim

Ā→∞ÿ�㕛

Ā�㕛=ā >0 → ∑ÿĀ, ∑ĀĀ both conv/div

b. lim

Ā→∞ÿ�㕛

Ā�㕛=0, ∑ĀĀ converges → ∑ÿĀ converges

c. lim

Ā→∞ÿ�㕛

Ā�㕛=∞, ∑ĀĀ diverges → ∑ÿĀ diverges

(6) Ratio Test

a. lim

Ā→∞ÿ�㕛+1

ÿ�㕛<1 → ∑ÿĀ converges

b. lim

Ā→∞ÿ�㕛+1

ÿ�㕛>1 or =∞ → ∑ÿĀ diverges

c. lim

Ā→∞ÿ�㕛+1

ÿ�㕛=1 → inconclusive (try other tests)

∑ÿĀ

Ā=1

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