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Engineering Maths flashcards
Fichas del conjunto (45)
- 1How does the Taylor Series expansion (T) approximate f(x) around the point a?The values of T and its derivatives are forced to be equal to f(x) and its derivatives at a.
- 2What is the single-variable form of Taylor series?
- 3What is the multi-variable form of Taylor series? (for f(x,y))
- 4What is the difference between a partial and total derivative?Only one variable is allowed to change when taking a partial derivative.
A total derivative takes into account the effect of all variables, but still with respect to only one. - 5You can only take the total derivative of a multi-variable function if...The other variables are also a function of the variable in question.
- 6How do you find the total derivative of a multi-variable function?Take the direct effect of the variable in question, plus the indirect effects of the others
- 7Given new coordinates for an integral, what steps must be taken to use them? (e.g. x = u/v & y = uv)Equate to find the new values of the bounds
Relate the area elements using by multiplying by the det Jacobian
Convert the function to be integrated - 8List methods of changing a problem to make it easier.Partial fractions
Change of coordinate system
Change of coordinates (substitution)
Change of domain - time, freq, phasor
Use of identities
Euler's formula - 9Why do we model waves using cosine rather than sine?Because cosine is the real term in Euler's formula.
- 10What is the transfer function (of a system)?An expression that relates the output of a system to its input signal.
- 11Describe how you convert a time-domain signal into a phasor one.Convert it to a complex exponential using Euler's formula and ignore the frequency/time dependent component (because a steady-state sinusoid is assumed)
- 12Describe how you convert a phasor signal into a time-domain one.Multiply by e(jωt) and take the real component.
- 13The s-plane is...The s-plane is the 'complex frequency' plane on which Laplace transforms are graphed, where... s= σ + jω
The real part of s represents the transient response. Without this, the transform decays to a phasor. - 14Describe the first steps you should take in multi-variable calculus.1) Look for geometric clues - e.g. equation of a circle
2) Draw the region as completely as possible - watch inner vs outer
3) Rearrange difficult limits - look for more geometric clues, check intersections - 15A point on a curve is a minimum if... A point on a curve is a point of inflexion if...Point of inflexion if the first non-zero derivative is of odd order.
- 16What is the significance of... 2 x y exp(x + y) = 0 ?No solutions for exp() = 0, so:
2 x y = 0 - 17How do you remember the orientation of transformation matrices?The variables on the rows of the output and transformation reflect each other.
- 18u(t) is used in Laplace to......ensure time domain signals
are zero for t < 0
It is assumed to be there if
not explicitly written - 19Differential current and voltage for a capacitor'Current - Capacitor'
- 20From first priciplesGo through full mathematical method without using tables etc.
- 21How to start Laplace transform (good practice)State 'F(s) = ' for easy referencing later
- 22Laplace is a l______ transformLaplace is a linear transform
- 23L[f'(t)]sF(s) - f(0)
- 24L[int(f(T) dT)]1/s F(s)
- 25How to apply Laplace to high order derivativesUse 'sF(s) - f(0)' iteratively
- 26Laplace impedance of a capacitor1/sC
- 27Laplace impedance of an inductorsL
- 28Laplace current through a capacitorI(s) = C[sV(s) - V(0)]
- 29Laplace voltage across a capacitorV(s) = 1/sC I(s) + V(0)/s
- 30Laplace current through an inductorI(s) = 1/sL V(s) + i(0)/s
- 31For functions of t, the variable s is always ________For functions of t, the variable s
is always constant - 32In Laplace, s is equal to: The real component affects: The imaginary component affects:s = σ + jω
Real - magnitude decay
Imaginary - frequency - 33If s is purely imaginary, a function's ________ is ________ Also...If s is purely imaginary,
a function's magnitude is constant
Also the laplace transform is then in phasor form - 34If a proof is not working...Work backwards
- 35Quotient rule(u/v)' = (u'v - uv')/v^2
- 36Jacobian matrix
- 37Hessian matrix
- 38Finding stationary points for multivariable functionsMust establish rate of change in every direction - select points where df/dx and df/dy are both zero
Take the discriminant - det(Hessian)
If Δ > 0 use second order of x to establish type, if Δ < 0 saddle point - 39Laplace magnitude of a (capacitor) voltage sourceV/s
- 40If asked to prove a complex differential relationship after a matrix transformation...Convert the relationships to operator form, then write the relationship to prove as nested operations
- 41—
- 42Laplace model of a capacitor
- 43Laplace model of an inductor
- 44Orientation of curvy capacitor symbol'Current pushes into plate'
- 45How to remember laplace 'current through' and 'voltage across' formulae1/sX OtherIV + initial(t)
Think about which component has which initial condition ie capacitor -> v(0) -> V(s) =
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- Mathematics Fichas sobre Engineering Maths flashcards, creado por Alexei Smith el 15/04/2018.
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- Alexei Smith
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