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University year 1 semester 2 > Mathematics 2 > Flashcards

Mathematics 2 Flashcards

1
Q

How is the partial derivative of f(x,y) w.r.t to y obtained?

A

Taking the derivative of f(x,y) w.r.t y, while holding x constant. y is kept constant when finding the derivative w.r.t x

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2
Q

How would you find the partial derivatives of f(x,y) = (x^2 + 4y).e^(-xy)

A

Use the product rule

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3
Q

If there is an equation with more than two variables, how would you find the partial derivatives?

A

Hold all other variables constant except the one differentiating w.r.t

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4
Q

What would be the partial derivative df/dx when f(x,y,z)=cos(xyz^2)?

A

-yz^2sin(xyz^2)

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5
Q

What is d^2f/dxdy and d^2f/dydx?

A 
d^2f/dxdy = d/dx(df/dy)
d^2f/dydx = d/dy(df/dx)
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6
Q

What is the slope in the direction at an angle of ‘a’ to the x-axis on a 3D surface given by?

A

(df/dx)cos(a) + (df/dy)sin(a) with df/dx and df/dy being partial derivatives

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7
Q

How is the total differential defined?

A

dz = (df/dx)dx + (df/dy)dy with df/dx and df/dy being partial derivatives

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8
Q

How is the maximum possible error estimated for V=(pi)(r^2)h?

A

dV=(dV/dr)dr + (dV/dh)dh = 2(pi)rh dr + (pi)(r^2) dh
Delta(V) = (dV/dr)delta(r) + (dV/dh)delta(h)
Delta(V) = |(dV/dr)||delta(r)| + |(dV/dh)||delta(h)|
Then substitute in values, with delta(variable) being the error in the variable and 2(pi)rh and (pi)(r^2) being the measured values of the variables.

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9
Q

What kind of ‘d’ is used for partial derivatives?

A

Curly ‘d’

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10
Q

z=f(x,y) where x and y are both functions of another variable t, such that x=x(t) and y=y(t). How is dz/dt found?

A

dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt) when dz/dx and dz/dy are partial derivatives.

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11
Q

How do you know when you have a maximum or minimum point on a 3D graph?

A

The partial derivatives, df/dx and df/dy at specified x and y points are equal to zero

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12
Q

On a 3D graph, how do you know if a stationary point is a saddle point or an extreme point? And if it is a maximum or minimum extreme point?

A

If D = (d^2f/dx^2)(d^2f/dy^2) - (d^2f/dxdy)^2 > 0 then it is an extreme point. If D is less than 0 then it is a saddle point. If d^2f/dx^2 > 0 or d^2f/dy^2 > 0 then it is a minimum, otherwise it’s a maximum. All the derivatives used here are partial derivatives, so use curly ‘d’s’.

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13
Q

How is the gradient of a 3D graph shown?

A

f = (df/dx)i + (df/dy)j
There is an upside down triangle with an arrow on top before ‘f’ and arrows above i and j. All arrows face right. All derivatives are partial derivatives.

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14
Q

Calculate gradient of function f(x,y) = x^2 + y^2

A

2xi + 2yj

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15
Q

How do you calculate multiple integrals?

A

Start by calculating the innermost integral and then work outwards.

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16
Q

How would you calculate the integral of xyz dxdydz?

A

Start by integrating with respect to x while holding y and z constant. Then integrate with respect to y while holding x and z constant. Then integrate with respect to z while holding x and y constant.

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17
Q

How do you calculate an integral of a function which is a product of functions, each of one variable only, for example f(x,y,z) = g(x).h(y).i(z)

A

The integral is calculated as a product of integrals, so the integral i(z).h(y).g(x)dxdydz = i(z)dz.h(y)dy.g(x)dx

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18
Q

In an m x n matrix, what are ‘m’ and ‘n’?

A 
m = number of rows
n = number of columns
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19
Q

What is a square matrix?

A

A matrix that has the same number of rows as columns (m = n) and it has order n.

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20
Q

What is the trace of a matrix?

A

The sum of all the elements on the main diagonal of the matrix.

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21
Q

What is a diagonal matrix?

A

A square matrix that has only it’s non-zero elements on the main diagonal.

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22
Q

What is the identity matrix I?

A

A diagonal matrix that has only ones on the main diagonal.

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23
Q

What is a zero or null matrix?

A

A matrix whose elements are all zero.

24
Q

What is the transpose of a matrix, for example matrix ‘A’. And how is it denoted.

A

The transpose of matrix A is denoted by A to the power of T and is obtained by swapping the rows and the columns.

25
Q

If matrix A is equal to the transpose of matrix A, what can be said about matrix A?

A

Matrix A is symmetric. The elements are symmetric about he main diagonal.

26
Q

If matrix A is equal to the minus of transpose of A, what can be said about matrix A?

A

Matrix A is skew-symmetric. A skew-symmetric matrix must have zero diagonal elements.

27
Q

How are matrices added and subtracted from one another?

A

If both matrices are m x n matrices then the same elements from each matrix are added/subtracted.

28
Q

How do you multiply a matrix by a scalar?

A

Each element in the matrix is multiplied by the scalar.

29
Q

How are matrices multiplied?

A

Each element of the new matrix is calculated by multiplying each element of the row of the first matrix by each element of the column of the second matrix, and summing these values.

30
Q

How is the determinant of matrix A denoted?

A

det A or |A|

31
Q

How is the determinant of a 2 x 2 matrix calculated?

A

The product of the main diagonal elements minus the product of the elements in the other diagonal.

32
Q

How do you calculate the determinant of a 3 x 3 matrix?

A

Take the first element in the first row, cover up the row and column that the element is in and calculate the determinant of the 2 x 2 matrix that is left, then multiply the determinant by the element that was chosen. Repeat for all the elements in the first row. Take away the product of the second element and the determinant and add the product of the third element and determinant.

33
Q

If any two rows or columns in a matrix are the same, then what does the determinant of the matrix equal?

A

0

34
Q

If any row or column in a matrix is a multiple of another, then what does the determinant equal?

A

0

35
Q

What is an adjoint matrix?

A

The transpose of the matrix of cofactors of A

36
Q

What is an inverse matrix?

A

A matrix that can be multiplied anyway with another matrix to give matrix I

37
Q

What is the equation for calculating the inverse matrix?

A

(1/|A|) x adj A

38
Q

When solving systems of equations using matrices, if Ax = b and we let Ai be the matrix A with the i-th column of A replaced by b, then how can the solution for the system of equations be found?

A

xi = |Ai|/|A|

39
Q

What is an augmented matrix?

A

A type of matrix used for the Gaussian method when the column vector b is added to the right hand end of the matrix A

40
Q

What are the main two steps for the Gaussian elimination?

A

1) by systematic combinations of row operations, one reduces the system of equations to ‘upper triangular’ form - where matrix A has 0’s under the main diagonal.
2) back substitution for solving the system of equations starting from the bottom right corner of the augmented ‘upper triangulated’ matrix. xn = (b’n/a’nn). Next, after solving for xn the equation in row n-1 becomes an equation with only one unknown, xn-1, which can be solved to give xn-1 = ((b’n-1 - (a’n-1,n . xn))/(a’n-1,n-1))

41
Q

What form of equation requires you to use the eigenvalue problem?

A

Ax = lambda x where lambda is an eigenvalue and x is an eigenvector

42
Q

How are eigenvalues found?

A

Ax = lambda x can be written as Ax = lambda Ix
Hence (A - lambda I)x = 0
To get non-trivial solutions of x, |A - lambda I| = 0
Expand |A - lambda I| to get a polynomial
Polynomial is solved to find values of lambda which are the eigenvalues.

43
Q

What is an ODE?

A

An equation that involves ordinary derivatives (a derivative of a function of one variable)

44
Q

What is the dependent variable and what is the independent variable?

A

The dependent variable is what we differentiate and the independent variable is the one we differentiate with respect to.

45
Q

What is the order of an ODE?

A

The degree of the highest derivative that is in the equation

46
Q

What is a linear ODE?

A

An ODE where the dependent variable and all it’s derivatives do not occur as products (x.(dx/dt)), raised to powers ((dx/dt)^2), or in any non linear function (trig, exp, log…)(e^x)

47
Q

What is a homogeneous ODE?

A

When all the terms involving the dependent variable (either itself or derivative) are on the left hand side of the equation and on the right hand side are the terms with the independent variable (or constants or zero), for a linear equation, if the right side of the equation is zero, the ODE is homogeneous.

48
Q

What is a separable equation?

A

All the ‘x’ terms can be put on one side and all the ‘t’ terms on the other side.

49
Q

What’s the general form of an exact differential equation?

A

p(x,t)(dx/dt) + q(x,t) = 0

50
Q

What’s the general form of a first order linear equation?

A

dx/dt + p(t)x = q(t) where p and a are just arbitrary functions of t only.

51
Q

How do you solve a first order linear equation?

A

Step 1 - generate integration factor, IF = e to the power of the integral of p(t) dt.
Step 2 - multiply equation through by IF. The left hand side is just (d/dt)(x times e to the power of the integral of p(t) dt)
Step 3 - integrate both sides with respect to t. Remember the constant.
Step 4 - get x on it’s own.

52
Q

When solving a second order homogeneous equation, what is the general solution when b^2 - 4ac is greater than zero, equal to zero, and less than zero?

A

When > 0,
x(t) = C1e^(lambda1 x t) + C2e^(lambda2 x t)
When = 0, x(t) = (C1 + C2t)e^(lambda x t)
When

53
Q

What is the general solution xg(t) of a linear non-homogeneous equation?

A

The sum of the general solution xh(t) to the homogeneous equation plus any specific specific solution xp(t) to non-homogeneous equation.
xg(t) = xh(t) = xp(t)

54
Q

If we have a pair of coupled first order ODEs for two unknowns, y1(t) and y2(t), where
y1’(t) = a11y1(t) + a12y2(t) and y2’(t) = a21y1(t) + a22y2(t), they can be written in matrix form as
y’ = Ay. Assume there’s a solution of the form
y = e^(lambda x t).X, the matrix equation turns into an eigenvalue problem. What is the general solution to the problem?

A 
y = e^(lambda1 x t).X1 and y = e^(lambda2 x t).X2 so the general solution is their sum:
y = A1e^(lambda1 x t).X1 + A2e^(lambda2 x t).X2
55
Q

When is a PDE linear?

A

When the unknown function f, and all partial derivatives appear only as themselves e.g. No products, multiplied at most by functions of the independent variables.

56
Q

When is a PDE homogeneous?

A

When every term involves the dependent variable (f) or it’s partial derivatives, otherwise it’s non-homogeneous.

57
Q

What is the rank of a matrix?

A

The number of non-zero rows when the matrix has been reduced to echelon form.

University year 1 semester 2 (4 decks)

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