Analysis of Several Variables (BMAT2), Odd Sem 2023

For a differentiable function \(f:\mathbb{R}^2\to \mathbb{R}\) we have \[f(x+h, y+k)= f(x, y) + hf_x(x,y) + kf_y(x,y)+\circ(\|(h,k)\|)\] The differential/ total differential of the function \(f\) is defined as \[ df(x,y) = \frac{\partial f}{\partial x}h + \frac{\partial f}{\partial y}k \] The differential is actually a function of \(x, y, h, k\) and represents the linear part of the difference \(f(x + h, y + k) - f(x, y)\). By considering the scalar fields \((x,y)\mapsto x\) and \((x,y)\mapsto y\) we can write \[ df(x, y) = f_x(x, y)dx + f_y(x, y)dy. \] If f has continuous partial derivatives of higher order we can compute the second differential \begin{align*} d^2f &= d(df) = h\frac{\partial}{\partial x}\left(h\frac{\partial f}{\partial x} + k\frac{\partial f}{\partial y} \right) + k\frac{\partial}{\partial y}\left(h\frac{\partial f}{\partial x} + k\frac{\partial f}{\partial y} \right) \\ &= f_{xx}(x,y)h^2 + 2f_{xy}(x,y)hk+f_{yy}(x,y)k^2 \\ &= f_{xx}(x,y)(dx)^2 + 2f_{xy}(x,y)(dx)(dy)+f_{yy}(x,y)(dy)^2 \end{align*} Note importantly that we have treated \(h\) and \(k\) as constants in the above computation. Often one writes \(dx^2\) instead of \((dx)^2\) but it is not to be confused with \(d(x^2)=2xdx\). Show that if higher order derivatives exist one can obtain \[ d^nf = \frac{\partial^n f}{\partial x^n} dx^n + n \frac{\partial^n f}{\partial x^{n-1}\partial y} dx^{n-1}dy + \dots + \frac{\partial^n f}{\partial y} \] which can be written symbolically as \[ \left( \frac{\partial}{\partial x} dx + \frac{\partial}{\partial y} dy \right)^n f. \] In interpreting the above, we expand the power and interpret terms like \( \left(\frac{\partial}{\partial x}dx \right)^n f\) as \( \left( \frac{\partial^n}{\partial x^n}f \right)(dx)^n\). Show that \[ f(x+h, y+k) = f(x,y) + df(x,y) + \frac{1}{2} d^2f(x,y) + \dots + \frac{1}{n!} d^nf(x,y) + R_n, \] where \[ R_n = \frac{1}{(n+1)!}d^{n+1}f(x + \theta h, y + \theta k), ~ 0 < \theta < 1. \] Work out the details in higher dimensions, when \(f:\mathbb{R}^n\to \mathbb{R}\).

• Find a vector \(V(x,y, z)\) normal to the surface \[ z = \sqrt{x^2+y^2} + (x^2 + y^2)^{3/2} \] at a general point \((x,y, z)\) of the surface, \((x,y,z)\ne 0 \).
• Find the cosine of the angle \(\theta\) between \(V(x,y,z)\) and the \(z\)-axis and determine the limit of \(\cos \theta\) as \((x,y,z)\mapsto (0,0,0)\).

Let \(f\) be defined on an open set \(S\) in \(\mathbb{R}^n\). We say that \(f\) is homogeneous of degree \(p\) over \(S\) if \(f(\lambda \mathbf{x}) = \lambda^p f(\mathbf{x})\) for every real \(\lambda\) and for every \(\mathbf{x}\) in \(S\) for which \(\lambda \mathbf{x}\in S\). If such a function differentiable at \(\mathbf{x}\), show that \[ \mathbf{x} \cdot \nabla f(\mathbf{x}) = p f(\mathbf{x}). \] Also prove the converse, that is, show that if \(\mathbf{x} \cdot \nabla f(\mathbf{x}) = p f(\mathbf{x})\) for all \(\mathbf{x}\) in an open set \(S\), then \(f\) must be homogeneous of degree \(p\) over \(S\).

Exercises 17, 18 of section 10.18 in Tom M. Apostol, Calculus Vol 2, second edition.

Let \(f\) and \(g\) be scalar fields with continuous first- and second-order partial derivatives on an open set \(S\) in the plane. Let \(\mathbf{R}\) denote a region (in \(S\)) whose boundary is a piecewise smooth Jordan curve \(C\). Prove the following identities, where \(\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \).

1. \[ \oint_{C} \frac{\partial g}{\partial n} \, \mathrm{d}s = \iint_{\mathbf{R}} \nabla^2g \; \mathrm{d}x\, \mathrm{d}y \]
2. \[ \oint_{C} f \frac{\partial g}{\partial n}\, \mathrm{d}s = \iint_{\mathbf{R}} (f\nabla^2g + \nabla f \nabla g)\; \mathrm{d}x\, \mathrm{d}y \]
3. \[ \oint_{C} \left( f \frac{\partial g}{\partial n} - g \frac{\partial f}{\partial n} \right)\,\mathrm{d}s = \iint_{\mathbf{R}} \left( f \nabla^2 g - g \nabla^2 f \right) \; \mathrm{d}x\, \mathrm{d}y \]

The identity c is known as Green's formula; it shows that \[ \oint_{C} f\frac{\partial g}{\partial n} \; \mathrm{d}s = \oint_{C} g \frac{\partial f}{\partial n}\; \mathrm{d}s \] whenever \(f\) and \(g\) are both harmonic on \(\mathbf{R}\) (that is, when \(\nabla^2 f= \nabla^2g = 0\) on \(\mathbf{R}\)).

Let \(I_k = \oint_{C_k} P dx + Q dy\), where \[ P(x,y) = -y \left[ \frac{1}{(x-1)^2 + y^2} + \frac{1}{x^2 + y^2} + \frac{1}{(x+1)^2 + y^2} \right] \] and \[ Q(x,y) = \frac{x-1}{(x-1)^2 + y^2} + \frac{x}{x^2 + y^2} + \frac{x + 1}{(x+1)^2 + y^2} \] In Figure 2.1, \(C_1\) is the smallest circle, \(x^2 + y^2 = \frac{1}{8}\) (traced clockwise), \(C_2\) is the largest circle, \(x^2 + y^2 = 4\) (traced counterclockwise), and \(C_3\) is the curve made up of the three intermediate circles \( (x -1)^2 + y^2 = \frac{1}{4}, \; x^2 + y^2 = \frac{1}{4}\) and \( (x +1)^2 + y^2 = \frac{1}{4} \) traced out as shown. If \(I_2 = 6 \pi \) and \( I_3 = 2 \pi \), find the value of \( I_1 \).

Let \(S\) be a parametric surface described by the explicit formula \(z = f(x,y)\), where \((x,y)\) varies over a plane region \(T\), the projection of \(S\) in the \(xy\)-plane. Let \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) and let \(\mathbf{n}\) denote the unit normal to \(S\) having a nonnegative \(z\)-component. Use the parametric representation \( \mathbf{r}(x,y) = x \mathbf{i} + y \mathbf{j} + f(x,y) \mathbf{k} \) and show that \[ \iint_{S} \mathbf{F}\cdot \mathbf{n} = \iint_{T} \left( -P \frac{\partial f}{\partial x} - Q \frac{\partial f}{\partial y} + R \right) \; \mathrm{d}x\, \mathrm{d}y, \] where each \( P, Q\) and \(R\) is to be evaluated at \( (x,y, f(x,y))\).

Let \(S\) be as in Problem 3, and \(\varphi\) be a scalar field. Show that

1. \[ \iint_{S} \varphi(x, y, z)\; \mathrm{d}S = \iint_{T} \varphi[x, y, f(x, y)] \sqrt{1 + \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2} \; \mathrm{d}x\, \mathrm{d}y. \]
2. \[ \iint_{S} \varphi(x, y, z)\; \mathrm{d}y \wedge \mathrm{d}z = - \iint_{T} \varphi[x, y, f(x, y)] \frac{\partial f}{\partial x}\; \mathrm{d}x\, \mathrm{d}y. \]
3. \[ \iint_{S} \varphi(x, y, z)\; \mathrm{d}z \wedge \mathrm{d}x = - \iint_{T} \varphi[x, y, f(x, y)] \frac{\partial f}{\partial y}\; \mathrm{d}x\, \mathrm{d}y. \]

Use Stokes' theorem to show that \[\oint (y^2+z^2) \mathrm{d}x + (x^2 + z^2) \mathrm{d}y + (x^2 + y^2) \mathrm{d}z = 2\pi ab^2, \] where \(C\) is the intersection of the hemisphere \(x^2 + y^2 + z^2 = 2ax, z > 0\), and the cylinder \(x^2 + y^2 = 2bx\), where \(0 < b < a\).

Let \(V\) be a convex region in \(3\)-space whose boundary is a closed surface \(S\) and let \(\mathbf{n}\) be the unit normal to \(S\). Let \(\mathbf{F}\) and \(\mathbf{G}\) be two continuously differentiable vector fields such that \(\text{curl}\, \mathbf{F} = \text{curl}\, \mathbf{G}\) and \(\text{div}\, \mathbf{F} = \text{div}\, \mathbf{G}\) everywhere in \(V\), and such that \(\mathbf{G}\cdot \mathbf{n} = \mathbf{F}\cdot \mathbf{n}\) everywhere on \(S\). Prove that \(\mathbf{F} = \mathbf{G}\) everywhere in \(V\).

[ Hint: Let \(\mathbf{H}= \mathbf{F}-\mathbf{G}\). Find a scalar field \(f\) such that \(\mathbf{H} = \nabla f\), and use a suitable identity to prove that \[\iiint_{V}\| \nabla f \|^2 \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z=0. \] From this deduce that \(\mathbf{H} = 0 \) in \(V\).]