Chapter 9

For each of the following 1 -forms \(\omega\), find if possible a function \(f\) such that \(\omega=d f\). (a) \(\omega=\left(3 x^{2} y+2 x y\right) d x+\left(x^{3}+x^{2}+2 y\right) d y\) (b) \(\omega=(x y \cos x y+\sin x y) d x+\left(x^{2} \cos x y+y^{2}\right) d y\) (c) \(\omega=\left(2 x y z^{3}+z\right) d x+x^{2} z^{3} d y+\left(3 x^{2} y z^{2}+x\right) d z\) \((d) \omega=x^{2} d y+3 x z d z\)

(a) If \(g\) is harmonic in \(\Omega\) and the normal derivative of \(g\) on the boundary is 0 , show that 1 \(\nabla g=0\) in \(\Omega\) (b) Let \(g\) and \(g^{*}\) be harmonic in \(\Omega\), and let \(\partial g / \partial \mathbf{n}=\partial g^{*} / \partial \mathbf{n}\) on \(\partial \Omega\). Show that \(g^{*}=g+K\) where \(K\) is constant.

Verify Green's theorem for \(\omega=x d x+x y d y\) with \(D\) as the unit square with opposite vertices at \((0,0),(1,1)\).

Let a \(=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}, \mathbf{b}=\mathbf{i}-\mathbf{j}+3 \mathbf{k}, \mathbf{c}=\mathbf{i}-2 \mathbf{j}\). Compute the vectors \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}, \mathbf{a} \times(\mathbf{b} \times \mathbf{c})\) \((a \times b) \times c, a \times(a \times b),(a+b) \times(b+c),(a \cdot b) c-(a \cdot c) b\)

Evaluate \(\int_{y}(x d x+x y d y)\) where (a) \(\gamma\) is the line \(x=t, y=t, 0 \leq t \leq 1\). (b) \(\gamma\) is the portion of the parabola \(y=x^{2}\) from \((0,0)\) to \((1,1)\). (c) \(\gamma\) is the portion of the parabola \(x=y^{2}\) from \((0,0)\) to \((1,1)\). (d) \(\gamma\) is the polygon whose successive vertices are \((0,0),(1,0),(0,1),(1,1)\).

Let \(\mathbf{V} \cdot \mathbf{F}=0\) in a convex region \(\Omega\). Show that \(\mathbf{F}\) can be expressed there in the form \(\mathbf{F}=\mathbf{\nabla} \times \mathbf{V}\), where \(\mathbf{\nabla} \cdot \mathbf{V}=0\) and \(\mathbf{\nabla}^{2} \mathbf{V}=-\mathbf{\nabla} \times \mathbf{F}\). (This reduces the problem of finding a 1 -form \(\omega\) with \(d \omega=\sigma\) for a given exact 2 -form \(\sigma\), to the solution of Poisson's equation.)

Apply Green's theorem to evaluate the integral of \(\left(x-y^{3}\right) d x+x^{3} d y\) ' around the circle \(x^{2}+y^{2}=1\)

If \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) are position vectors, show that the vector \(\mathbf{n}=\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}\) is a normal to the plane through the points \(\mathrm{a}, \mathrm{b}, \mathrm{c}\).

Evaluate \(\int_{y}(y d x-x d y)\) where (a) \(\gamma\) is the closed curve \(x=t^{2}-1, y=t^{3}-t,-1 \leq t \leq 1\) (b) \(\gamma\) is the straight line from \((0,0)\) to \((2,4)\) (c) \(\gamma\) is the portion of the parabola \(y=x^{2}\) from \((0,0)\) to \((2,4)\). (d) \(\gamma\) is the polygon whose successive vertices are \((0,0),(-2,0),(-2,4),(2,4)\).

Verify Stokes' theorem with \(\omega=x d z\) and with \(\Sigma\) as the surface described by \(x=u v\) \(y=u+t ; z=u^{2}+r^{2}\) for \((u, t)\) in the triangle with vertices \((0,0),(1,0),(1,1)\)

Three points \(p_{j}=\left(x_{j}, y_{j}, z_{j}\right)\) which do not lie in a plane through the origin determine a trihedral with sides \(\overrightarrow{0 p_{1}}, \overrightarrow{\overrightarrow{b p}_{2}}, \overrightarrow{0 p_{3}}\) which has positive orientation if and only if $$ \left|\begin{array}{lll} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{array}\right|>0 $$ Using this, show that the vectors a, b, and \(\mathrm{a} \times \mathbf{b}\) form a trihedral having positive orientation. unless a and \(\mathrm{b}\) are parallel.

Consider the differential form $$ \omega=\frac{x d x+y d y}{x^{2}+y^{2}} $$ Show that \(d \omega=0\) in the ring \(D .\) Is \(\omega\) exact in \(D ?\)

Evaluate \(\int_{y}\left(z d x+x^{2} d y+y d z\right)\) where (a) \(\gamma\) is the straight line from \((0,0,0)\) to \((1,1,1)\). (b) \(\gamma\) is the portion of the twisted cubic \(x=t, y=t^{2}, z=t^{3}\) from \((0,0,0)\) to \((1,1,1)\). (c) \(\gamma\) is the portion of the helix \(x=\cos t, y=\sin t, z=t\), for \(0 \leq t \leq 2 \pi\) (d) \(\gamma\) is the closed polygon whose successive vertices are \((0,0,0),(2,0,0),(2,3,0),(0,0,1)\). \((0,0,0)\)

Given vectors a and \(\mathbf{b}\), and a real number \(k\), when is there a vector \(v\) such that \(\mathbf{a} \times \mathbf{v}=\mathbf{b}\) and \(\mathbf{a} \cdot \mathrm{v}=k ?\)

Recall that a function \(f\) is said to be homogeneous of degree \(k\) if $$ f(x t, y t, z t)=t^{k} f(x, y, z) $$ for all \(t \geq 0\) and all \((x, y, z)\) in a sphere about the origin. Let $$ \omega=A d x+B d y+C d z $$ be an exact 1 -form whose coefficuents are all homogeneous of degree \(k, k \geq 0 .\) Show that \(\omega=d f\), where $$ f(x, y, z)=\frac{x A(x, y, z)+y B(x, y, z)+z C(x, y, z)}{k+1} $$

Prove the divergence theorem directly when \(R\) is the solid sphere $$ x^{2}+y^{2}+z^{2} \leq 1 $$

If such exist, find integrating factors for the following differential forms: (a) \(\left(x^{2}+2 y\right) d x-x d y\) (b) \(3 y z^{2} d x+x z^{2} d y+2 x y z d z\) (c) \(x y d x+x y d y+y z d z\)

If \(\omega=B(y) d y\), show that \(\int_{\gamma} \omega=\int_{y_{0}}^{y_{1}} B(y) d y\) for any smooth curve \(y\) which starts at \(\left(x_{0}, y_{0}\right)\) and ends at \(\left(x_{1}, y_{1}\right)\).

(a) Show that the area of a region \(D\) to which Green's theorem applies may be given by $$ A(D)=\frac{1}{2} \int_{\partial D}(x d y-y d x) $$ (b) Apply this to find the area bounded by the ellipse \(x=a \cos \theta, y=b \sin \theta, 0 \leq \theta \leq 2 \pi\).

Verify the following: (a) \((3 x d x+4 y d y)\left(3 x^{2} d x-d y\right)=-\left(3 x+12 x^{2} y\right) d x d y\) (b) \(\left(3 x^{2} d x-d y\right)(3 x d x+4 y d y)=\left(3 x+12 x^{2} y\right) d x d y\). (c) \((x d y-y z d z)(y d x+x y d y-z d z)=\left(x y^{2} z-x z\right) d y d z-y^{2} z d z d x-x y d x d y\). (d) \(\left(x^{2} d y d z+y z d x d y\right)(3 d x-d z)=\left(3 x^{2}-y z\right) d x d y d z\). (e) \((d x d y-d y d z)(d x+d y+d z)=0 .\) (f) \((d x-x d y+y z d z)\left(x d x-x^{2} d y+x y z d z\right)=0 .\)

Let \(\sigma=A d y d z+B d z d x+C d x d y\), where the functions \(A, B\), and \(C\) are homogeneous of degree \(k\) in a neighborhood of the origin. If \(\sigma\) is exact, show that \(\sigma=d \omega\), where $$ \omega=\frac{(z B-y C) d x+(x C-z A) d y+(y A-x B) d z}{k+2} $$

Show that $$ \begin{aligned} (A d x&+B d y+C d z)(a d x+b d y+c d z) \\ &=\left|\begin{array}{ll} B & C \\ b & c \end{array}\right| d y d z+\left|\begin{array}{ll} C & A \\ c & a \end{array}\right| d z d x+\left|\begin{array}{ll} A & B \\ a & b \end{array}\right| d x d y \end{aligned} $$

Show that the following 2 -forms are exact by exhibiting each in the form \(\sigma=d \omega\) : (a) \(\left(3 y^{2} z-3 x z^{2}\right) d y d z+x^{2} y d z d x+\left(z^{3}-x^{2} z\right) d x d y\) (b) \((2 x z+z) d z d x+y d x d y\)

Show that $$ (A d x+B d y+C d z)(a d y d z+b d z d x+c d x d y)=(a A+b B+c C) d x d y d z $$

Show that the volume of a suitably well-behaved region \(R\) in space is given by the formula $$ \mathrm{v}(R)=\frac{1}{3} \iint_{i R} x d y d z+y d z d x+z d x d y $$

Let \(p=(x, y, z)\) and \(r-|p| .\) Find the gradient of the functions \(r^{2}, r, 1 / r, r^{m}\), and \(\log r\).

Formulate a necessary condition that a 1 -form in \(n\) variables be exact.

Evaluate \(d f\) if \((a) f(x, y, z)=x^{2} y z ;(b) f(x, y)=\log \left(x^{2}+y^{2}\right)\)

Show that the gradient vectors for \(f\) are orthogonal to the level surfaces $$ f(x, y, z)=c $$

Verify the invariance relation \((d \omega)^{*}=d\left(\omega^{*}\right)\) when \(\omega=x d y d z\) and \(T\) is the transformation \(x=u+v-w, y=u^{2}-v^{2}, z=v+w^{2}\)

Let \(A\) be the square matrix \(\left[a_{i j}\right]\) and let \(B\) be a nonsingular matrix. Set \(A^{*}=B^{-1} A B\). Show that the trace of \(A\) (the sum of the diagonal entries) is the same as that of \(A^{*}\).

Formulate a definition for "integrating factor" for 2 -forms. Obtain a differential equation which must be satisfied by any integrating factor for the 2 -form $$ \sigma=A d y d z+B d z d x+C d x d y $$ Using this, and Euler's differential equation for homogeneous functions (Exercise 11, Sec. 3.4), obtain the most general integrating factor for $$ x d y d z+y d z d x+z d x d y $$

Evaluate \(d \omega\) where (a) \(\omega=x^{2} y d x-y z d z\) (b) \(\omega=3 x d x+4 x y d y\) (c) \(\omega=2 x y d x+x^{2} d y\) (d) \(\omega=e^{x y} d x-x^{2} y d y\) (e) \(\omega=x^{2} y d y d z-x z d x d y\) \((f) \omega=x^{2} z d y d z+y^{2} z d z d x-x y^{2} d x d y\) \((g) \omega=x z d y d x+x y d z d x+2 y z d y d z\)

Assuming Green's theorem for rectangles, prove it for a region of the type described by \((9-29)\) with \(\omega=A(x, y) d x\) by means of the transformation \(x=u, y=f(u) v+g(u)(1-r)\) \(a \leq u \leq b, 0 \leq v \leq 1\)

Let \(\Omega\) be a region in space which can be mapped onto a star-shaped set by a 1-to-1 transformation of class \(C^{\prime \prime} .\) Show that any 2 -form \(\sigma\) which satisfies the equation \(d \sigma=0\) in \(\Omega\) is exact in \(\Omega\).

$$ \text { Verify }(9-24) $$

Verify the following calculations with forms in four variables: (a) \((d x+d y-d z+d w)(d x d y+2 d z d w)\) \(=2 d x d z d w+2 d y d z d w-d x d y d z+d x d y d w\) : (b) \((d x d y+d y d z-d z d w)(x d x d y+y d z d w)=(y-x) d x d y d z d w\). (c) \(d\left(x^{2} y d y d w+y w^{2} d x d z+x y z w d x d y\right)=\left(x y w-w^{2}\right) d x d y d z\) \(+(2 x y+x y z) d x d y d w+2 y w d x d z d w\) (d) \(d x d y d z d w=\frac{c(x \cdot y, z \cdot w)}{r(r, s, t, u)} d r d s d t d u\) when \(x=\phi(r . s, t, u), y=\psi(r, s, t, u), z=\theta(r, s, t, u), w=q(r, s, t, u)\)

$$ \text { Verify }(9-25) $$

Evaluate \(\iint_{\Sigma}(x y d y d z+y=d x d w)\), where \(\Sigma\) is the two- dimensional surface in 4 -space described by \(x=r^{2}+s^{2}, y=r-s, z=r s, w=r+s\), and \((r, s)\) obeys \(0 \leq r \leq 1,0 \leq s \leq 1\).

$$ \text { Show that div }(\operatorname{grad} f \times \operatorname{grad} g)=0 . $$