Applied Mathematics-I
- How are complex numbers used to analyse AC electrical circuits with impedance and phasors? In one line In AC circuits a sinusoidal voltage or current is represented by a complex number (phasor), and each component by a complex impe... 1
- How do I find real and imaginary parts of inverse circular functions like tan⁻¹(α + iβ)? In one line Write the inverse function as an ordinary equation, e.g. tan⁻¹(α+iβ)=x+iy means tan(x+iy)=α+iβ , then use the conjugate and sepa... 1
- What is a partial derivative of a function of two or three variables and what does it mean? In one line A partial derivative measures how a function of several variables changes when only one variable changes and all the others are... 1
- What are higher-order and mixed partial derivatives, and why is u_xy usually equal to u_yx? In one line Differentiating a partial derivative again gives second-order derivatives u_xx,u_yy and the mixed ones u_xy,u_yx , which are equ... 1
- How do I differentiate a composite function z = f(x, y) when x and y depend on t? In one line If z=f(x,y) with x=x(t) and y=y(t) , then dz/dt=∂ z/∂ xdx/dt+∂ z/∂ ydy/dt : add the contributions of each path. Explained simply... 1
- How do I prove results like (∂/∂x + ∂/∂y + ∂/∂z)²u = −9/(x+y+z)² for a given function? In one line Compute the needed partial derivatives, combine them as the question asks, simplify using algebraic identities, and only then ap... 1
- How do I use the chain rule when x and y depend on two new variables u and v? In one line If z=f(x,y) with x=x(u,v) , y=y(u,v) , then z_u=z_xx_u+z_yy_u and z_v=z_xx_v+z_yy_v ; every path from z to the new variable cont... 1
- What is a homogeneous function and how do I find its degree? In one line A function f(x,y) is homogeneous of degree n if scaling every variable by t scales the function by tⁿ : f(tx,ty)=tⁿf(x,y) . Expl... 1
- What is a total differential and how is it used to estimate small changes and errors? In one line For u=f(x,y) , the total differential du=∂ u/∂ xdx+∂ u/∂ ydy gives the approximate change in u when x and y change by small amou... 1
- What is Euler's theorem on homogeneous functions of two variables and how is it proved? In one line If u is a homogeneous function of degree n in x and y , then x∂ u/∂ x+y∂ u/∂ y=nu . Explained simply Go back to the rectangle: a... 1
- How do I find dy/dx for an implicit function f(x, y) = 0 using partial derivatives? In one line If y is defined implicitly by f(x,y)=0 , then dy/dx=-f_x/f_y , provided f_y≠0 . Explained simply Some curves, like a circle x²+y... 1
- What is the second-order deduction of Euler's theorem, x²u_xx + 2xy u_xy + y²u_yy = n(n−1)u? In one line For a homogeneous function u of degree n , x²u_xx+2xyu_xy+y²u_yy=n(n-1)u ; it follows from differentiating Euler's theorem once... 1
- How does Euler's theorem extend to homogeneous functions of three variables x, y and z? In one line If u(x,y,z) is homogeneous of degree n , then xu_x+yu_y+zu_z=nu ; it is the same theorem with one more term. Explained simply Th... 1
- How do I apply Euler's theorem when f(u), not u, is homogeneous, like u = sin⁻¹((x+y)/(√x+√y))? In one line If f(u) is homogeneous of degree n , then xu_x+yu_y=nf(u)/f'(u)=G(u) and x²u_xx+2xyu_xy+y²u_yy=G(u)[G'(u)-1] . Explained simply... 1
- What are the Cartesian, polar and exponential forms of a complex number and how do I convert between them? In one line A complex number can be written as x+iy (Cartesian), r(cosθ+isinθ) (polar) or re^iθ (exponential); all three describe the same p... 1
- How do I add, multiply and divide complex numbers and use modulus, argument and conjugate? In one line Complex numbers follow ordinary algebra with the single extra rule i²=-1 ; the conjugate bar z=x-iy is the tool that removes i f... 1
- What does De Moivre's theorem state and how does it simplify powers of cos θ + i sin θ? In one line De Moivre's theorem says (cosθ+isinθ)ⁿ=cos nθ+isin nθ : raising this number to the power n simply multiplies the angle by n . Ex... 1
- How do I expand cos nθ and sin nθ in powers of cos θ and sin θ? In one line Write cos nθ+isin nθ=(cosθ+isinθ)ⁿ , expand the right side by the binomial theorem, and compare real and imaginary parts. Explai... 1
- How do I express powers like cosⁿθ and sinⁿθ in terms of cosines and sines of multiples of θ? In one line Put x=cosθ+isinθ , so 2cosθ=x+1/x and 2isinθ=x-1/x ; raise to the power, expand, and pair terms into x^k±frac1x^k . Explained si... 1
- Show that the minimum value of \(u=xy+a^3\left(\frac1x+\frac1y\right)\) is \(3a^2\). u_x=y-a³/x², u_y=x-a³/y² . Stationary points: x²y=a³,\ xy²=a³ . Dividing, x=y ; then x³=a³⇒ x=y=a . Point (a,a) . r=u_xx=2a³/x³,\ s=u_xy=1,\... 1
- If \(y=a\cos(\log x)+b\sin(\log x)\), prove that \(x^2y_{n+2}+(2n+1)xy_{n+1}+(n^2+1)y_n=0\). y₁=-a/xsin(log x)+b/xcos(log x)⇒ xy₁=-asin(log x)+bcos(log x) . Differentiate: xy₂+y₁=-1/x[acos(log x)+bsin(log x)]=-y/x⇒ x²y₂+xy₁+y=0 . Lei... 1
- Expand \(\sin^5\theta\cos^3\theta\) in a series of sines of multiples of \(\theta\). Let x=cosθ+isinθ : x-1/x=2isinθ,\ x+1/x=2cosθ,\ xⁿ-frac1xⁿ=2isin nθ . (2isinθ)⁵(2cosθ)³=(x-1/x)⁵(x+1/x)³=(x-1/x)²(x²-frac1x²)³ =(x²-2+frac1x... 1
- If \(\cos\alpha+\cos\beta+\cos\gamma=0,\ \sin\alpha+\sin\beta+\sin\gamma=0\), prove that (i) \(\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta+\gamma)\) (ii) \(\sin3\alpha+\sin3\beta+\sin3\gamma=3\sin(\alpha+\beta+\gamma)\) Let a=cosα+isinα,\ b=cosβ+isinβ,\ c=cosγ+isinγ . Then a+b+c=(cosα+cosβ+cosγ)+i(sinα+sinβ+sinγ)=0 . Identity: if a+b+c=0 then a³+b³+c³=3abc (... 1
- Find all the roots of \(\left(\frac12+i\frac{\sqrt3}{2}\right)^{3/4}\), hence find the continued product of the roots. Step 1 (polar form): 1/2+isqrt3/2=cosπ/3+isinπ/3 . Including the general angle, 1/2+isqrt3/2=cos(2kπ+π/3)+isin(2kπ+π/3) . Step 2 (De Moivre)... 1