Let \(f:\R^n\to\R^m\) and \(a\in\R^n\). What does “\(f\) is differentiable at \(a\)” mean?
Define the directional derivative of \(f\) at \(a\) in the direction of a vector \(v\in\R^n\).
Let \(f:\R^n\to\R\) be differentiable at \(a\in\R^n\). Define the gradient \(\nabla f(a)\).
Define \(F:\R^2\to\R^2\) by \(F(x,y)=(x^2-xy,\; x+y^2)\). Compute \(DF(x,y)\) and use it to write the “best” linear approximation of \(F\) at \((1,2)\).
Let \(A\) be an \(m\times n\) matrix and \(b\in\R^m\). Define \(F:\R^n\to\R^m\) by \(F(x)=Ax+b\). Compute the derivative of \(F\).
Let \(f,g:\R^n\to\R^m\) be differentiable at \(a\in\R^n\). Define \(h:\R^n\to\R\) by \(h(x)=f(x)\cdot g(x)\), i.e., \(h(x)\) is defined to be the dot product of \(f(x)\) and \(g(x)\).
Compute \(Dh\) in terms of \(f\) and \(g\) and \(Df\) and \(Dg\).
Recall that for \( \displaystyle\begin{pmatrix}x&y\\ z&w\end{pmatrix}\) we have \(\displaystyle\det\begin{pmatrix}x&y\\ z&w\end{pmatrix}=xw-yz\).
Let \( I=\displaystyle\begin{pmatrix}1&0\\0&1\end{pmatrix}\) and \(H=\displaystyle\begin{pmatrix}a&b\\ c&d\end{pmatrix}\).
Compute the derivative of \(\det\) at \(I\) in the direction \(H\).
Let \(\mathrm{inv}:GL_2(\R)\to M_2(\R)\) be the map sending a \(2\)-by-\(2\) matrix to its inverse.
Fix \(H\in M_2(\R)\). Compute the derivative of the inverse map at the identity \(I\) in the direction \(H\).
Define \(g:\R^2\to\R\) by \[ g(x,y)= \begin{cases} \dfrac{x^2y}{x^4+y^2} & (x,y)\neq(0,0), \\ 0 & (x,y)=(0,0). \end{cases} \] For nonzero \(v \in \R^2\), compute the directional derivative \(D_v g(0,0)\).
Is the function \(g\) in S01P08 differentiable at \((0,0)\)?
Define \(f:\R^2\to\R\) by \(f(x,y)=\sqrt{1+x^2+y^2}\).
Show that for all \((x,y) \in \R^2\), \[ 0\le f(x,y)-1 \le \frac12(x^2+y^2). \]
Use this fact to bound the error of a linear approximation to \(f\) at \((0,0)\).
Finally find an explicit value of \(\epsilon\) so that \(1 - \epsilon < \sqrt{1+0.01^2+0.02^2} < 1 + \epsilon\).
Let \(f:\R^n\to\R^m\) be differentiable at \(a\). Prove that \(f\) is continuous at \(a\).
Let \(f,g:\R^n\to\R^m\) be differentiable at \(a\in\R^n\). Prove from the definition of differentiability that \(f+g\) is differentiable at \(a\) and that \[ D(f+g)(a)=Df(a)+Dg(a). \]
Let \(f:\R^n\to\R\) be differentiable at \(a\) with gradient \(\nabla f(a)\neq 0\).
Prove that among all unit vectors \(u\), the maximum of \(D_u f(a)\) equals \(\|\nabla f(a)\|\).
Then show that this maximum occurs in the direction \(u=\nabla f(a)/\|\nabla f(a)\|\).
Let \(f:\C\to\C\) and write \(z=x+iy\). Suppose \[ f(z)=u(x,y)+iv(x,y) \] with \(u,v:\R^2\to\R\).
What does it mean to say that \(f\) is complex differentiable at \(z_0=x_0+iy_0\)?
When \(f\) is complex differentiable, verify that the Cauchy–Riemann equations hold at \((x_0,y_0)\), i.e., show that \[ u_x(x_0,y_0)=v_y(x_0,y_0),\qquad u_y(x_0,y_0)=-v_x(x_0,y_0). \]
For the following problems, first decide whether the statement is true or false as written. If it is true, give a complete proof that works in full generality (not just examples!) and make clear where the hypotheses are used. If it is false, disprove it with a counterexample: give a specific example that satisfies the assumptions but violates the conclusion. Justify that your example is actually a counterexample! Then try to salvage the statement by making a minimal, natural change—that could be adding a missing hypothesis or weakening the conclusion. State your corrected version clearly and prove it.
If all partial derivatives of \(f:\R^2\to\R\) exist at \(a\), then \(f\) is differentiable at \(a\).
If for \(f:\R^n\to\R\) all directional derivatives \(D_u f(a)\) exist when \(\|u\|=1\), and the map \(u\mapsto D_u f(a)\) is continuous on the unit sphere, then \(f\) is differentiable at \(a\).