Mathematical Methods UPDATED ACTUAL
QUESTIONS AND CORRECT ANSWERS
hermitian conjugate - Correct answers✔✔The transpose of the complex conjugate
infinite dimensional vectors as functions - Correct answers✔✔A vector assigns each index value
a number. For N dimensional there are N such index values.
A function f(x) assigns to each value of x its argument. There are infinitely many values of this
argument and in this sense is an infinite dimensional function.
Dirichlet Conditions - Correct answers✔✔1. function must be periodic
2. it must be single values and continuous except at finite number within each period of finite
discontinuities
3. finite number of extrema in one period
4. integral over one period |f(x)| must converge.
properties of conservative fields - Correct answers✔✔1. can be expressed as the gradient of a
scalar field
2. its curl vanishes
3. Its path integral only depends on the start and end points.
divergence theorem - Correct answers✔✔States net flux flux of a vector through a closed surface
around a given volume is the same as the divergence of same function within enclosed surface V.
Can be seen as a conservation law: amount of vector field flowing out the volume is given by
sources minus sinks of this volume.
, convolution - Correct answers✔✔If f(x) spatially dependent signal to be measured and g(x)
describes the resolution function of detector then measurement will be given by the convolution
of signal and resolution. Relevant in photography.
Fourier Series - Correct answers✔✔Fourier series express periodic functions in terms of their
frequency components. Examples include signal compressing by removing the higher order
frequencies before storage.
Laplace Transform - Correct answers✔✔Generally used to solve differential equations as
differential operators in laplace space. Used in electrical circuits where they simplify the
equations.
Solenoidal - Correct answers✔✔divergence equals 0
Irrotational - Correct answers✔✔curl equals 0
Gibbs Phenomenon - Correct answers✔✔first few terms of fourier series overshoot the value for
small n
For larger n the overshoot does not disappear but the position of each overshoot has moved
closer to the discontinuity.
harmonic - Correct answers✔✔when the grad squared of a function equals zero
conversion between coordinate systems - Correct answers✔✔reexpressing the position in the
new coordinates
converting the unit vectors
for differential operators: expressing the operators in the new basis system (may affect unit
vectors)
QUESTIONS AND CORRECT ANSWERS
hermitian conjugate - Correct answers✔✔The transpose of the complex conjugate
infinite dimensional vectors as functions - Correct answers✔✔A vector assigns each index value
a number. For N dimensional there are N such index values.
A function f(x) assigns to each value of x its argument. There are infinitely many values of this
argument and in this sense is an infinite dimensional function.
Dirichlet Conditions - Correct answers✔✔1. function must be periodic
2. it must be single values and continuous except at finite number within each period of finite
discontinuities
3. finite number of extrema in one period
4. integral over one period |f(x)| must converge.
properties of conservative fields - Correct answers✔✔1. can be expressed as the gradient of a
scalar field
2. its curl vanishes
3. Its path integral only depends on the start and end points.
divergence theorem - Correct answers✔✔States net flux flux of a vector through a closed surface
around a given volume is the same as the divergence of same function within enclosed surface V.
Can be seen as a conservation law: amount of vector field flowing out the volume is given by
sources minus sinks of this volume.
, convolution - Correct answers✔✔If f(x) spatially dependent signal to be measured and g(x)
describes the resolution function of detector then measurement will be given by the convolution
of signal and resolution. Relevant in photography.
Fourier Series - Correct answers✔✔Fourier series express periodic functions in terms of their
frequency components. Examples include signal compressing by removing the higher order
frequencies before storage.
Laplace Transform - Correct answers✔✔Generally used to solve differential equations as
differential operators in laplace space. Used in electrical circuits where they simplify the
equations.
Solenoidal - Correct answers✔✔divergence equals 0
Irrotational - Correct answers✔✔curl equals 0
Gibbs Phenomenon - Correct answers✔✔first few terms of fourier series overshoot the value for
small n
For larger n the overshoot does not disappear but the position of each overshoot has moved
closer to the discontinuity.
harmonic - Correct answers✔✔when the grad squared of a function equals zero
conversion between coordinate systems - Correct answers✔✔reexpressing the position in the
new coordinates
converting the unit vectors
for differential operators: expressing the operators in the new basis system (may affect unit
vectors)
