Differentiability of Scalar Fields

Question 1

Define

Answer 1

Question 2

Define continuous.

Answer 2

Question 3

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

Answer 3

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

Question 4

Define an open ball.

Answer 4

Question 5

Define an open set.

Answer 5

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

Question 6

Define a neighbourhood.

Answer 6

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

Question 7

Define a closed set.

Answer 7

Question 8

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

Answer 8

Need to do

Question 9

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

Answer 9

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

Question 10

Finish this definition.

Answer 10

Question 11

What is v(a) a generalisation of?

Answer 11

df/dx in 1-dimension

Question 12

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

Answer 12

Question 13

What is the theorem that links partial derivatives and differentiabilty?

Answer 13

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.

Question 14

Define continuously differentiable.

Answer 14

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

Question 15

Define continuously differentiable on an open set U.

Answer 15

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

Question 16

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.

Answer 16

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

Question 17

What is a smooth fucntion?

Answer 17

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

Question 18

Finish the following theorem.

Answer 18

Question 19

What formula can you use to show something is differentiable?

Answer 19

Where v(a) equals _∇_f

Question 20

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

Answer 20

Question 21

Define a level set.

Answer 21

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.

Question 22

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

Answer 22

Level curve

Question 23

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

Answer 23

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

Question 24

Define an explicit function.

Answer 24

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

Question 25

Define an **implicit funciton**.

Answer 25

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

Question 26

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

Answer 26

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

Question 27

What is the implicit function theorem?

Answer 27

Question 28

What is a critical value?

Answer 28

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

Question 29

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

Answer 29

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

Question 30

Prove

Answer 30

Question 31

Finish the following.

Answer 31

Question 32

Prove the following.

Answer 32

Question 33

What is parametric form for a normal line?

Answer 33

Question 34

What is the equation for the tangent plane?

Answer 34

Question 35

Define **interior**.

Answer 35

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

Question 36

Define **global maximum.**

Answer 36

Question 37

Define **global minimum.**

Answer 37

Question 38

Define **strict global maximum.**

Answer 38

Question 39

Define **local maximum point.**

Answer 39

Question 40

Define a **strict local maximum** point.

Answer 40

Question 41

What is another name for maximum and minimum points?

Answer 41

Extremal points

Question 42

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

Answer 42

Extremal values

Question 43

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

Answer 43

Critical

Question 44

Are all critical points local maxima/minima?

Answer 44

Nope

Question 45

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

Answer 45

Saddle point

Question 46

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_)?

Answer 46

Question 47

What does _h_._∇_ equal in the following?

Answer 47

Question 48

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

Answer 48

Question 49

What is H called in the following equation?

Answer 49

The Hessian Matrix

Question 50

What is the Hessian Matrix?

Answer 50

Question 51

How do you write the Hessian matrix in index form?

Answer 51

Question 52

Why is the Hessian symmetric?

Answer 52

Because f_xy = f_yx

Question 53

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

Answer 53

Question 54

Why is the sign of _h_^TH**_h_** important?

Answer 54

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

Question 55

What critical point at _a_ do we have if _h_^TH**_h_** \> 0?

Answer 55

For all small _h_, _a_ will be a strict local minimum

Question 56

What critical point do we have if _h_^TH_h​_ \< 0?

Answer 56

For all small _h_, _a_ will be strict local maximum

Question 57

Define **positive definite.**

Answer 57

Question 58

Define **negative definite**.

Answer 58

Question 59

Define **indefinite**.

Answer 59

Question 60

Define **positive semi-definite.**

Answer 60

Question 61

Define **negative semi-definite.**

Answer 61

Question 62

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

Answer 62

It is a strict local minimum

Question 63

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

Answer 63

It is a strict lcoal maximum

Question 64

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

Answer 64

It is a saddle point

Question 65

What are three tests for positive definiteness?

Answer 65

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

Question 66

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

Answer 66

The second derivative test for functions of 2 variables.

Question 67

Describe the second derivative test for functions of 2 varibales.

Answer 67

Question 68

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

Answer 68

Question 69

Define **bounded.**

Answer 69

Question 70

What is the extreme value theorem?

Answer 70

Question 71

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

Answer 71

Question 72

What method do you require in the following problem?

Answer 72

Largrange multipliers

Question 73

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?

Answer 73

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_

Question 74

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

Answer 74

Question 75

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

Answer 75

Question 76

What is λ known as?

Answer 76

The Lagrange multiplier

Question 77

How can you rewrite the following by introducing the Lagrangian?

Answer 77

Question 78

What is the equation for the Lagrangian?

Answer 78

Question 79

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

Answer 79

Define the Lagrangian, then use simultaneous equations to find an _a_ such that _∇_fL(_a_) = _0_