Partial Differentiation
- 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
- 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
- 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 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
- 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
- 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 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
- 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 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 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
- 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
- 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
- If \(u=\sin^{-1}\left(\frac{x+y}{\sqrt x+\sqrt y}\right)\), prove that \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=-\frac{\sin u\cos2u}{4\cos^3u}\) Given: u=sin⁻¹(x+y/sqrt x+sqrt y)⇒ z=sin u=x+y/sqrt x+sqrt y . Step 1 (homogeneity): z(tx,ty)=t(x+y)/sqrt t(sqrt x+sqrt y)=t^(1/2)z , so z=s... 1
- If \(u=2(ax+by)^2-k(x^2+y^2)\) and \(a^2+b^2=k\), find \(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\). ∂ u/∂ x=4a(ax+by)-2kx⇒∂²u/∂ x²=4a²-2k ∂ u/∂ y=4b(ax+by)-2ky⇒∂²u/∂ y²=4b²-2k ∂²u/∂ x²+∂²u/∂ y²=4(a²+b²)-4k=4k-4k ∂²u/∂ x²+∂²u/∂ y²=0 1
- If \(u=f\left(e^{x-y},e^{y-z},e^{z-x}\right)\), then prove that \(\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z}=0\). Let X=e^(x-y),\ Y=e^(y-z),\ Z=e^(z-x) , so u=f(X,Y,Z) . ∂ X/∂ x=X,\ ∂ X/∂ y=-X,\ ∂ X/∂ z=0 ; ∂ Y/∂ x=0,\ ∂ Y/∂ y=Y,\ ∂ Y/∂ z=-Y ; ∂ Z/∂ x=-Z... 1
- If \(u=\frac{\sqrt x+\sqrt y}{x+y}\) find \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}\). u(tx,ty)=sqrt t(sqrt x+sqrt y)/t(x+y)=t^(-1/2)u , so u is homogeneous of degree n=-1/2 . By Euler's theorem xu_x+yu_y=nu : x∂ u/∂ x+y∂ u/∂ y... 1
- If \(z=f(x,y),\ x=r\cos\theta,\ y=r\sin\theta\) then prove that \(\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2=\left(\frac{\partial z}{\partial r}\right)^2+\frac1{r^2}\left(\frac{\partial z}{\partial\theta… ∂ x/∂ r=cosθ,\ ∂ y/∂ r=sinθ,\ ∂ x/∂θ=-rsinθ,\ ∂ y/∂θ=rcosθ . Chain rule: ∂ z/∂ r=∂ z/∂ xcosθ+∂ z/∂ ysinθ ...(1) ∂ z/∂θ=-rsinθ∂ z/∂ x+rcosθ∂... 1
- For \(u=\sin^{-1}(x^2+y^2)^{1/5}\) deduce \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial y^2}=\frac2{25}\tan u\,(2\tan^2u-3)\). z=sin u=(x²+y²)^(1/5) is homogeneous of degree n=2/5 . Euler: xz_x+yz_y=nz⇒cos u (xu_x+yu_y)=2/5sin u⇒ xu_x+yu_y=2/5tan u=g(u) . Deduction:... 1
- If \(u=\log(x^3+y^3+z^3-3xyz)\), prove that \(\left(\frac\partial{\partial x}+\frac\partial{\partial y}+\frac\partial{\partial z}\right)^2u=-\frac9{(x+y+z)^2}\). u_x=3x²-3yz/x³+y³+z³-3xyz,\ u_y=3y²-3zx/·s,\ u_z=3z²-3xy/·s u_x+u_y+u_z=3(x²+y²+z²-xy-yz-zx)/(x+y+z)(x²+y²+z²-xy-yz-zx)=frac3x+y+z (using x³... 1
- If \(u=(1-2xy+y^2)^{-1/2}\), prove that \(x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=y^2u^3\). ∂ u/∂ x=-1/2(1-2xy+y²)^(-3/2)(-2y)=y u³ ∂ u/∂ y=-1/2(1-2xy+y²)^(-3/2)(-2x+2y)=(x-y) u³ x∂ u/∂ x-y∂ u/∂ y=xy u³-(xy-y²)u³ =y²u³ (proved). 1
- If \(H=f(y-z,\,z-x,\,x-y)\), then prove that \(\frac{\partial H}{\partial x}+\frac{\partial H}{\partial y}+\frac{\partial H}{\partial z}=0\). Let u=y-z,\ v=z-x,\ w=x-y , so H=f(u,v,w) . ∂ u/∂ x=0,\ ∂ v/∂ x=-1,\ ∂ w/∂ x=1 ; ∂ u/∂ y=1,\ ∂ v/∂ y=0,\ ∂ w/∂ y=-1 ; ∂ u/∂ z=-1,\ ∂ v/∂ z=1... 1
- If \(u=\sin^{-1}\left(\frac{x+y}{\sqrt x+\sqrt y}\right)\), then prove that \(x^2\frac{\partial^2u}{\partial x^2}+2xy\frac{\partial^2u}{\partial x\partial y}+y^2\frac{\partial^2u}{\partial x^2}=-\frac{\sin u\cos2u}{4\cos^3u}\). (Note: the last term on the left is y²∂²u/∂ y² ; it is misprinted as ∂ x² in the paper.) Given: u=sin⁻¹(x+y/sqrt x+sqrt y)⇒ z=sin u=x+y/sqrt... 1
- State & prove Euler's theorem on homogeneous functions with two independent variables. Statement: If u=f(x,y) is a homogeneous function of degree n in x and y, then x∂ u/∂ x+y∂ u/∂ y=nu . Proof: Since u is homogeneous of degree... 1