Question 3.137

Show that the equation \(x+y-z+\cos(xyz)=0\) can be solved for \(z=g(x, y)\) near the origin. Find \(\displaystyle \frac{\partial g}{\partial x}\) and \(\displaystyle \frac{\partial g}{\partial y}\) at (0, 0).

Question 3.138

Show that \(xy + z + 3xz^5 =4\) is solvable for \(z\) as a function of \((x,y)\) near (1, 0, 1). Compute \(\partial z/ \partial x\) and \(\partial z/\partial y\) at (1, 0).

Question 3.139

• (a) Check directly (i.e., without using Theorem 11) where we can solve the equation \(F(x,y)=y^2 + y + 3x +1 = 0\) for \(y\) in terms of \(x\).
• (b) Check that your answer in part (a) agrees with the answer you expect from the implicit function theorem. Compute \(dy/dx\).

Question 3.140

Repeat Exercise 3 with \(F(x,y)= xy^2 -2y + x^2 +2 = 0\).

Question 3.141

Let \(F(x,y)=0\) define a curve in the \(xy\) plane through the point \((x_0,y_0)\), where \(F\) is \(C^1\). Assume that \((\partial F/\partial y)\) \((x_0,y_0)\neq 0\). Show that this curve can be locally represented by the graph of a function \(y=g(x)\). Show that (i) the line orthogonal to \({\nabla}\! F(x_0,y_0)\) agrees with (ii) the tangent line to the graph of \(y=g(x)\).

Question 3.142

Consider the surface \(S\) given by \(3y^2z^2-3x=0\).

• (a) Using the implicit function theorem, verify that we can solve for \(x\) as a function of \(y\) and \(z\) near any point on \(S\). Explicitly write \(x\) as a function of \(y\) and \(z\).
• (b) Show that near (1, 1, \(-1\)) we can solve for either \(y\) or \(z\), and give explicit expressions for these variables in terms of the other two.

Question 3.143

Show that \(x^3 z^2 - z^3 yx =0\) is solvable for \(z\) as a function of \((x,y)\) near (1, 1, 1), but not near the origin. Compute \(\partial z/\partial x\) and \(\partial z/\partial y\) at (1, 1).

Question 3.144

Discuss the solvability in the system \begin{eqnarray*} 3x + 2y + z^2 + u + v^2 &=& 0\\ 4x + 3y + z + u^2 + v + w + 2 &=& 0\\ x+ z + w + u^2 +2 &=& 0 \end{eqnarray*} for \(u,v,w\) in terms of \(x,y,z\) near \(x=y=z=0,u=v=0,\) and \(w=-2\).

Question 3.145

Discuss the solvability of \begin{eqnarray*} y + x + uv &=& 0\\ uxy + v &=& 0 \end{eqnarray*} for \(u, v\) in terms of \(x,y\) near \(x=y=u=v=0\) and check directly.

Question 3.146

Investigate whether or not the system \begin{eqnarray*} u(x,y,z) &=& x+ xyz\\ v(x,y,z) &=& y+ xy\\ w(x,y,z) &=& z + 2x + 3z^2 \end{eqnarray*} can be solved for \(x,y,z\) in terms of \(u,v,w\) near \((x,y,z) = (0,0,0)\).

Question 3.147

Consider \(f(x,y) = ((x^2 - y^2)/(x^2 + y^2)\), \(xy/(x^2 + y^2))\). Does this map of \({\mathbb R}^2 \backslash (0,0)\) to \({\mathbb R}^2\) have a local inverse near \((x,y) = (0,1)\)?

Question 3.148

• (a) Define \(x{:}\,\, {\mathbb R}^2 \rightarrow {\mathbb R}\) by \(x(r,\theta) = r\cos \theta\) and define \(y{:}\, \,{\mathbb R}^2\rightarrow {\mathbb R}\) by \(y(r,\theta) = r\sin \theta\). Show that \[ \frac{\partial (x,y)}{\partial (r,\theta)}\Big| _{(r_0,\theta_0)} = r_0. \]
• (b) When can we form a smooth inverse function \((r(x,y),\theta (x,y))\)? Check directly and with the inverse function theorem.
• (c) Consider the following transformations for spherical coordinates (see Section 1.4): \begin{eqnarray*} x(\rho, \phi,\theta) &=& \rho \sin \phi \cos \theta\\ x(\rho, \phi,\theta) &=& \rho \sin \phi \sin \theta \\ z(\rho, \phi,\theta) &=& \rho \cos \phi. \end{eqnarray*} Show that the Jacobian determinant is given by \[ \frac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} = \rho^2 \sin \phi. \]
• (d) When can we solve for \((\rho,\phi,\theta)\) in terms of \((x,y,z)\)?

Question 3.149

Let \((x_0,y_0,z_0)\) be a point of the locus defined by \(z^2+xy -a =0, z^2 + x^2 - y^2 -b =0\), where \(a\) and \(b\) are constants.

• (a) Under what conditions may the part of the locus near \((x_0,y_0,z_0)\) be represented in the form \(x=f(z), y= g(z)\)?
• (b) Compute \(f'(z)\) and \(g'(z)\).

Question 3.150

Consider the unit sphere \(S\) given by \(x^2+y^2+z^2=1\). \(S\) intersects the \(x\) axis at two points. Which variables can we solve for at these points? What about the points of intersection of \(S\) with the \(y\) and \(z\) axes?

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Question 3.151

Let \(F(x, y)=x^3-y^2\) and let \(C\) denote the level curve given by \(F(x, y)=0\).

• (a) Without using the implicit function theorem, show that we can describe \(C\) as the graph of \(x\) as a function of \(y\) near any point.
• (b) Show that \(F_x(0,0)=0\). Does this contradict the implicit function theorem?

Question 3.152

Consider the system of equations \begin{eqnarray*} x^5v^2+2y^3u&=&3\\ 3yu-xuv^3&=&2. \end{eqnarray*} Show that near the point \((x, y, u, v)= (1, 1, 1, 1)\), this system defines \(u\) and \(v\) implicitly as functions of \(x\) and \(y\). For such local functions \(u\) and \(v\), define the local function \(f\) by \(f(x, y)=(u(x, y), v(x, y))\). Find \(Df(1, 1)\).

Question 3.153

Consider the equations \begin{eqnarray*} x^2-y^2-u^3+v^2+4&=&0\\ 2xy+y^2-2u^2+3v^4+8&=&0. \end{eqnarray*}

• (a) Show that these equations determine functions \(u(x, y)\) and \(v(x, y)\) near the point \((x, y, u, v)=(2, -1, 2, 1)\).
• (b) Compute \(\frac{\partial u}{\partial x}\) at \((x, y)=(2, -1)\).

Question 3.154

Is it possible to solve the system of equations \begin{eqnarray*} xy^2 + xzu + yv^2 &=& 3\\ u^3 yz + 2xv - u^2 v^2 &=& 2 \end{eqnarray*} for \(u(x,y,z), v(x,y,z)\) near \((x,y,z)= (1,1,1), (u,v) = (1,1)\)? Compute \(\partial v/\partial y\) at \((x,y,z)=(1, 1, 1)\).

Question 3.155

The problem of factoring a polynomial \(x^n + a_{n-1} x^{n-1}+ \cdots + a_0\) into linear factors is, in a sense, an “inverse function” problem. The coefficients \(a_i\) may be thought of as functions of the \(n\) roots \(r_j\). We would like to find the roots as functions of the coefficients in some region. With \(n=3\), apply the inverse function theorem to this problem and state what it tells you about the possibility of doing this.