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AMV - Michaelmas > Differentiability of Scalar Fields > Flashcards

Differentiability of Scalar Fields Flashcards

1
Q

Define

A
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2
Q

Define continuous.

A
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3
Q

Finish this theorem: If f and g are continuous at a then so are … ?

A
  1. f + g
  2. fg
  3. f/g provided that g(a) ≠ 0
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4
Q

Define an open ball.

A
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5
Q

Define an open set.

A

A subset S of ℝn is open (S is an open set) if for each point a ∈ S there is an open ball Bδ(a) which is also in S.

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6
Q

Define a neighbourhood.

A

A neighbourhood N os a point a ∈ ℝn is a subset of ℝn which includes an open set containing a.

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7
Q

Define a closed set.

A
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8
Q

Prove every open ball Bδ(a) in ℝn is open .

A

Need to do

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9
Q

Define continuous on U (U is an open subset).

A

If U is an open subset of ℝn and f:U ➝ ℝ is a function then f is said to be continuous on U if it is continuous at each point in U

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10
Q

Finish this definition.

A
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11
Q

What is v(a) a generalisation of?

A

df/dx in 1-dimension

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12
Q

If v exists prove it is equal to _∇_f.

A
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13
Q

What is the theorem that links partial derivatives and differentiabilty?

A

Let f:U ➝ ℝ be a function on an open set U⊂ℝn, and suppose a ∈ U. If all partial derivativs of f exist and are continuous in a neighbourhood of a, then f is differentiable at a.

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14
Q

Define continuously differentiable.

A

A function is continuously differentiable at a if it and all of its partial derivatives exist and are continuous at a

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15
Q

Define continuously differentiable on an open set U.

A

A function is continuosuly differentiable on an open set U if it and all of its partial derivatives exist and are continuous on U.

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16
Q

What does the following theorem imply if a function is continuosuly differentiable on an open set? And why is this important?

Let f:U ➝ ℝ be a function on an open set U⊂ℝn, and suppose a ∈ U. If all partial derivativs of f exist and are continuous in a neighbourhood of a, then f is differentiable at a.

A

It is also differentiable there - important because continuous differentiability is easier to check than differentiability.

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17
Q

What is a smooth fucntion?

A

A function in which all partial derivatives of all orders exist.

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18
Q

Finish the following theorem.

A
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19
Q

What formula can you use to show something is differentiable?

A

Where v(a) equals _∇_f

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20
Q

If f(x): U ➝ ℝ is differentiable with U an open set in ℝn, what is the chain rule ?

A
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21
Q

Define a level set.

A

A level set S of f:U ➝ ℝ, where U is an open subset of ℝn, is the set of points {x ∈ U : f(x) = c} for some constant c.

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22
Q

What is the level set usually called when n=2?

A

Level curve

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23
Q

How to do you find a level set when f(x,y) =c?

A

Set the function equal to c and then rearrange to make y the subject.

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24
Q

Define an explicit function.

A

y = g(x) gives y an an explicit function of x

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25
Q

Define an implicit funciton.

A

f(x,y) = 0 gives y as an implicit function of x.

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26
Q

How do you go from an explicit to implicit funciton where g(x) = y?

A

Set f(x,y) = g(x) - y

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27
Q

What is the implicit function theorem?

A
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28
Q

What is a critical value?

A

The point Q on a level curve f = c at which ∇ f = 0, the value of c is called a critical value.

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29
Q

How can’t the level curve be written at a critical value Q?

A

The level curve cannot be written either as y = g(x) or as x = h(y) in the neighbourhood of Q

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30
Q

Prove

A
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31
Q

Finish the following.

A
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32
Q

Prove the following.

A
33
Q

What is parametric form for a normal line?

A
34
Q

What is the equation for the tangent plane?

A
35
Q

Define interior.

A

A point a ∈ X is in the interior of X, a ∈ int(X) is there is an open ball Bδ(a), for some δ > 0, which is also in X.

36
Q

Define global maximum.

A
37
Q

Define global minimum.

A
38
Q

Define strict global maximum.

A
39
Q

Define local maximum point.

A
40
Q

Define a strict local maximum point.

A
41
Q

What is another name for maximum and minimum points?

A

Extremal points

42
Q

What are the values taken at maximum and minimum points called?

A

Extremal values

43
Q

Local extreme of differentiable fucntions occur at what sort of points?

A

Critical

44
Q

Are all critical points local maxima/minima?

A

Nope

45
Q

What is a critical point called that is neither a max or min?

A

Saddle point

46
Q

Suppose f(x, y) and all partial derivatives up to order n + 1 are continuous in some neighbourhood a. Then for h = (h1, h2) small we have, with a = (x0, y0) what Taylor series expansion do we have for f(a + h)?

A
47
Q

What does h.∇ equal in the following?

A
48
Q

If a is a critical point what does the following become?

A
49
Q

What is H called in the following equation?

A

The Hessian Matrix

50
Q

What is the Hessian Matrix?

A
51
Q

How do you write the Hessian matrix in index form?

A
52
Q

Why is the Hessian symmetric?

A

Because fxy = fyx

53
Q

Neglecting higher terms of the taylor expansion what does the following equal?

A
54
Q

Why is the sign of hTHh important?

A

It tells you if the point a is a local maximum or minimum.

55
Q

What critical point at a do we have if hTHh > 0?

A

For all small h, a will be a strict local minimum

56
Q

What critical point do we have if hTH_h​_ < 0?

A

For all small h, a will be strict local maximum

57
Q

Define positive definite.

A
58
Q

Define negative definite.

A
59
Q

Define indefinite.

A
60
Q

Define positive semi-definite.

A
61
Q

Define negative semi-definite.

A
62
Q

If H (the hessian matrix) is positive definite what does this say about the point a?

A

It is a strict local minimum

63
Q

If H (the hessian matrix) is negative definite what does this say about the point a?

A

It is a strict lcoal maximum

64
Q

If H (the hessian matrix) is indefinite what does this say about the point a?

A

It is a saddle point

65
Q

What are three tests for positive definiteness?

A
  1. Find the eigenvalues of H and refer to the poitns about their values
  2. H is positive definite if the determinatn of all n of its upper-left sqaure submatrices is positive
  3. Second derivative test for functions of 2 variables
66
Q

Which test is the one you should use to test for positive definiteness?

A

The second derivative test for functions of 2 variables.

67
Q

Describe the second derivative test for functions of 2 varibales.

A
68
Q

What is the theorem about that links global extemum and local extremum.

A
69
Q

Define bounded.

A
70
Q

What is the extreme value theorem?

A
71
Q

What are the four steps to find global extrema of f in a closed and bounded (compact) region X?

A
72
Q

What method do you require in the following problem?

A

Largrange multipliers

73
Q

Suppose f and g are scalar functions on ℝ2 with continuous partial derivatives and that the contours of f are f(x,y) = d. There is also no critical points of f on the level set g(x,y) = 0 so it is a smooth curve. As we move along g(x,y) = 0 through some point (x0, y0) when is the value of f constant?

A
  1. If we are following a contour line of f, i.e. contours of f and g are parallel at (x0,y0)
  2. f id anyway constant at that point, i.e. _∇_f(x0, y0) = 0
74
Q

How do you check if you are following a contour line?

A
75
Q

What are the solutions at a = (x0, y0) if you are following a contout line of f?

A
76
Q

What is λ known as?

A

The Lagrange multiplier

77
Q

How can you rewrite the following by introducing the Lagrangian?

A
78
Q

What is the equation for the Lagrangian?

A
79
Q

How could you find the max and min values of a function on a curve?

A

Define the Lagrangian, then use simultaneous equations to find an a such that _∇_fL(a) = 0

AMV - Michaelmas (8 decks)

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