17.3 Partial and total derivatives, differentials[2D3]

E396

[1FX]Check that the following partial derivatives exist, and compute them:

\begin{align*} \frac{\partial ~ }{\partial {x}} {\Big( 4xy + 3x^ 2 y -z y^ 2 \Big)}~ ~ & ,~ ~ \frac{\partial ~ }{\partial {y}} \Big( 4xy + 3x^ 2 y -z y^ 2 \Big)\\ \frac{\partial ~ }{\partial {x}} { \frac{ze^{x+|y|}}{1+x^ 2|y|} }~ ~ & ,~ ~ \frac{\partial ~ }{\partial {z}} \frac{ze^{x+|y|}}{1+x^ 2|y|} \end{align*}

Hidden solution: [UNACCESSIBLE UUID ’1FY’]

E396

[1FZ] Prerequisites:Riemann integral,2.Let \(I⊆ ℝ\) open interval with \(0∈ I\). Given \(f=f(x,y):I×[0,1]→ℝ\) continuous, and such that also \(\frac{∂ }{∂ x} f\) exists and is continuous, set

\[ g(x)=∫_ 0^ 1 f(x,y) \, {\mathbb {d}}y\quad , \]

show that \(g\) is of class \(C^ 1\), and that

\[ g'(x)=∫_ 0^ 1 \frac{\partial ~ }{\partial {x}} f(x,y) \, {\mathbb {d}}y~ . \]

Hidden solution: [UNACCESSIBLE UUID ’1G0’][UNACCESSIBLE UUID ’1G1’]

E396

[1G2] Prerequisites:Riemann integral, 6, 6, 2.Let

\[ h(t) = ∫_{a(t)}^{b(t)} f(t,z) \, {\mathbb {d}}z \]

where \(a,b,f\) are \(C^ 1\) class functions: show that \(h\) is class \(C^ 1\) and calculate the derivative.

Hidden solution: [UNACCESSIBLE UUID ’1G3’]

E396

[1G4]Are the following functions differentiable in \((0,0)\)?

\begin{align*} f_ 1(x,y)= \begin{cases} x+y & \text{if}~ x{\gt}0\\ x+ye^{-x^ 2} & \text{if}~ x≤ 0 \end{cases}& ,& f_ 2(x,y) = \sqrt{x^ 2+y^ 2} \\ f_ 3(x,y) = \left( \arctan (y+1)\right)^{x+1} & ,& f_ 4(x,y) =\max \{ x^ 2, y^ 2\} ~ ~ . \end{align*}

Hidden solution: [UNACCESSIBLE UUID ’1G5’]

E396

[1G6]Prerequisites:1.Let \(f:ℝ^ k→ℝ\) be of class \(C^∞\). Recall that, by Schwarz’s theorem, permutiation of the order of partial derivatives does not change the result. Let \(N(n,k)\) be the number of partial (potentially different) derivatives of order \(n\): show that \(N(n,k)=\binom {n+k-1}{k-1}\) (which is a polynomial with integer coefficients in the variable \(n\), of order \(k-1\)). Hidden solution: [UNACCESSIBLE UUID ’1G7’]

E396

[1G8] Let \(W⊆ ℝ^ n\) be an open nonempty set, fix \(\overline x∈ W\). Let then \(𝜓:W→ℝ\) of class \(C^ 2\). Let \(∇𝜓(\overline x)\) be the row vector of coordinates \(\frac{\partial ~ }{\partial {x_ k}} 𝜓(\overline x)\) (which is the gradient of \(𝜓\), a special case of the ”Jacobian matrix”); we abbreviate it to \(D=∇𝜓(\overline x)\) for simplicity; let \(H\) be the Hessian matrix of components \(H_{h,k}= \frac{\partial {}^ 2}{\partial {x_ k x_ h}}𝜓 (\overline x) \); show the validity of Taylor’s formula at the second order

\[ 𝜓 (\overline x+v) = 𝜓 (\overline x) + D v + \frac 1 2 v^ t H v + o(|v|^ 2) \]

(note that the product \(D v\) is a matrix \(1× 1\) that we identify with a real number, and similarly for \(v^ t H v\)).

[UNACCESSIBLE UUID ’1G9’]

[1GB] Prerequisites:6.Let \(V,W⊆ ℝ^ n\) be open nonempty sets, and \(G:V→ W\) of class \(C^ 2\). Fix \(\overline y∈ V\) and \(\overline x=G(\overline y)∈ W\). Suppose that \(𝜓:W→ℝ\) is of class \(C^ 2\); define \(\tilde𝜓 = 𝜓 ◦ G\), then compare Taylor’s second-order formulas for \(𝜓\) and \(\tilde𝜓\) (centered in \(\overline x\) and \(\overline y\), respectively). Assuming also that \(G\) is a diffeomorphism, verify that

• \(\overline x\) is a stationary point for \(𝜓\) if and only if \(\overline y\) is stationary point for \(\tilde𝜓\),

• and in this case the Hessians of \(𝜓\) and \(\tilde𝜓\) are similar (i.e. the matrices are equal, up to coordinate changes).

Hidden solution: [UNACCESSIBLE UUID ’1GC’]