RepozytoriumzprojektemPython/stopy/Ćwiczenia_7.ipynb

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   "source": [
    "# Ćwiczenia  7"
   ]
  },
  {
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   "id": "1276423e",
   "metadata": {},
   "source": [
    "***TEMAT:*** funkcje wielu zmiennych - gradient, płaszczyzna styczna, reguła łańcucha, pochodne cząstkowe wyższych rzędów"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adcbd788",
   "metadata": {},
   "source": [
    "## Gradient, płaszczyzna styczna i reguła łańcucha"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da9cb4e5-acae-4d5b-85a9-f86d1ec751f8",
   "metadata": {},
   "source": [
    "### Gradient"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "516f7bfe-4937-4093-9d5a-69b53ba527df",
   "metadata": {},
   "source": [
    "Gradientem funkcji $f$ w punkcie $(x_1,x_2,...,x_n)$, w którym istnieją wszystkie pochodne cząstkowe funkcji $f$, nazywamy wektor\n",
    "\n",
    "$$\n",
    "\\nabla f(x_1,x_2,...,x_n) = \\left(\\frac{\\partial f}{\\partial x_1}(x_1,x_2,...,x_n), ...,  \\frac{\\partial f}{\\partial x_n}(x_1,x_2,...,x_n)\\right).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aefc1880-8ee7-4884-a7d9-833f27ad016b",
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   "source": [
    "#### Zadanie 1\n",
    "\n",
    "Wyznacz gradient następujących funkcji:\n",
    "1. $f(x,y)=x^2+3xy$\n",
    "2. $g(x,y,z)=ze^{xy}$\n",
    "\n",
    "Jaka jest interpretacja tego wektora?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d24ad98e-0737-437f-9e08-8e2a5e8a1d26",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62192ddd-704b-4754-8f86-3c3ffda39ad0",
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   "source": [
    "Mamy\n",
    "$$\n",
    "\\nabla f(x,y)=\\left(2x+3y,3x\\right)\n",
    "$$\n",
    "oraz\n",
    "$$\n",
    "\\nabla g(x,y,z)=\\left(yze^{xy},xze^{xy},e^{xy}\\right).\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33d09d4f-c4af-40e3-bcbb-4f916dd5f611",
   "metadata": {},
   "source": [
    "___Interpretacja geometryczna gradientu___\n",
    "\n",
    ">1. Gradient funkcji w punkcie wskazuje __kierunek  najszybszego  wzrostu  funkcji  w  tym punkcie.__\n",
    ">\n",
    ">2. Gradient funkcji w punkcie jest __prostopadły  do poziomicy  funkcji__ przechodzącej przez ten punkt."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98578fbf-a1df-4d3b-b8d8-dc4d0009c722",
   "metadata": {},
   "source": [
    "### Płaszczyzna styczna do powierzchni"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd8b2261-a2a4-4394-bdaf-e07257516e62",
   "metadata": {},
   "source": [
    "Jeśli funkcja $f(x,y)$ posiada w punkcie $(x_0,y_0)$ pochodne cząstkowe, to płaszczyznę zadaną równaniem\n",
    "$$ \n",
    "\\frac{\\partial f}{\\partial x} (x_0, y_0)(x-x_0) + \\frac{\\partial f}{\\partial y}(x_0, y_0)(y-y_0)-(z-z_0)= 0 .\n",
    "$$\n",
    "nazywamy płaszczyzną styczną do wykresu funkcji $f$ w punkcie  $(x_0, y_0,f(x_0, y_0))$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06be3815-19e7-442e-b873-c0b190ebb60e",
   "metadata": {},
   "source": [
    "#### Zadanie 2\n",
    "\n",
    "Napisz wzór płaszczyzny stycznej do wykresu funkcji $f(x,y)=x^2+y^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "537b3a74-e291-4e66-97be-323f7ff344b2",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:"
   ]
  },
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   "cell_type": "markdown",
   "id": "f58c17e2-0d2b-4c07-9fa8-5165000dd0ed",
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   "source": [
    "Płaszczyzna styczna do wykresu naszej funkcji w punkcie $(x_0,y_0,x_0^2+y_0^2)$ ma równanie\n",
    "$$\n",
    "2x_0(x-x_0)+2y_0(y-y_0)-(z-x_0^2-y_0^2)=0\n",
    "$$\n",
    "Dla $(x_0,y_0)=(0,0)$ daje to równanie\n",
    "$$\n",
    "z=0\n",
    "$$\n",
    "a dla $(x_0,y_0)=(1,1)$ daje to równanie\n",
    "$$\n",
    "2x+2y-z-2=0.\n",
    "$$\n"
   ]
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   "outputs": [
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    "import IPython.display as display\n",
    "from IPython.display import IFrame\n",
    "IFrame('https://www.geogebra.org/calculator/ahx8sscp', width=800, height=600, style=\"border: 1px solid black\")"
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   "cell_type": "markdown",
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   "metadata": {},
   "source": [
    "### Reguła łańcucha "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2eb1c13-3ce9-46e3-9cb7-21cf7d74c8d2",
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   "source": [
    "Niech $f,g,h\\colon \\mathbb{R}^2 \\to \\mathbb{R}$ będą funkcjami dwóch zmiennych rzeczywistych o wartościach rzeczywistych. Jeżeli:\n",
    "\n",
    "1. $g,h$ mają w punkcie $(x_0,y_0)$ pochodne cząstkowe, oraz\n",
    "\n",
    "2. $f(u,v)$ ma w punkcie $(u_0,v_0)$ pochodne cząstkowe, gdzie $u=g(x_0,y_0),v=h(x_0,y_0)$, \n",
    "\n",
    "to funkcja złożona $F(x,y)=f(g(x,y),h(x,y))$ ma w punkcie $(x_0,y_0)$ pochodne cząstkowe równe:\n",
    "$$\n",
    "\\frac{\\partial F}{\\partial x}(x_0,y_0)=\\frac{\\partial f}{\\partial u}(u_0,v_0)\n",
    "\\cdot \\frac{\\partial g}{\\partial x}(x_0,y_0)+\\frac{\\partial f}{\\partial v}(u_0,v_0)\n",
    "\\cdot \\frac{\\partial h}{\\partial x}(x_0,y_0)\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\frac{\\partial F}{\\partial y}(x_0,y_0)=\\frac{\\partial f}{\\partial u}(u_0,v_0)\n",
    "\\cdot \\frac{\\partial g}{\\partial y}(x_0,y_0)+\\frac{\\partial f}{\\partial v}(u_0,v_0)\n",
    "\\cdot \\frac{\\partial h}{\\partial y}(x_0,y_0)\n",
    "$$"
   ]
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  {
   "cell_type": "markdown",
   "id": "d7587290-fc47-469a-b378-c79a372f4157",
   "metadata": {},
   "source": [
    "#### Zadanie 3\n",
    "Niech \n",
    "$$\n",
    "f(x,y)=\\sin(x^2+y),\\quad g(x,y)=x^2+y\\quad \\text{ oraz } \\quad h(x,y)=xe^y.\n",
    "$$\n",
    "Obliczymy pochodne cząstkowe funkcji \n",
    "$$ \n",
    "F(x,y)=f(g(x,y),h(x,y)).\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6447f860-e4ec-400b-8d71-b7582ba6530b",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ea9e169-4457-4d4c-8804-ff50ade4f495",
   "metadata": {},
   "source": [
    "__I sposób__\n",
    "\n",
    "Ze wzorów wynika,  że\n",
    "$$\n",
    "F(x,y)=\\sin(x^4+2x^2y+y^2+xe^{y}).\n",
    "$$\n",
    "Zatem \n",
    "$$\n",
    "\\frac{\\partial F}{\\partial x}(x,y)=\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot(4x^3+4xy+e^y)\n",
    "$$\n",
    "oraz\n",
    "$$\n",
    "\\frac{\\partial F}{\\partial y}(x,y)=\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot(2x^2+2y+xe^y).\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebd66e96-50fd-44ce-8afe-375b26fc0e7d",
   "metadata": {},
   "source": [
    "__II sposób__\n",
    "\n",
    "Skorzystamy tym razem z reguły łańcucha.\n",
    "\n",
    "Mamy\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial u}(u,v)= 2u\\cos(u^2+v)\\quad{ oraz }\\quad \n",
    "\\frac{\\partial f}{\\partial v}(u,v)=\\cos(u^2+v).$$\n",
    "\n",
    "Zatem \n",
    "$$\n",
    "\\frac{\\partial F}{\\partial x}(x,y)=2(x^2+y)\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot 2x+\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot e^y=\n",
    "\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot(4x^3+4xy+e^y).\n",
    "$$\n",
    "oraz\n",
    "$$\n",
    "\\frac{\\partial F}{\\partial y}(x,y)=2(x^2+y)\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot 1+\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot xe^y=\n",
    "\\cos(x^4+2x^2y+y^2+xe^{y})\\cdot(2x^2+2y+xe^y).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6e27613-9dc9-44f5-80ae-9d5730e75baf",
   "metadata": {},
   "source": [
    "##  Pochodne cząstkowe wyższych rzędów"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9edc2c8f-97d8-4052-b231-5c5a718e9af8",
   "metadata": {},
   "source": [
    "#### Zadanie 4\n",
    "Wyznacz wszystkie pochodne cząstkowe rzędu drugiego dla funkcji \n",
    "$$f(x,y)=e^{x^2+y}.$$\n",
    "Co ciekawego zauważasz?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab762d49-d4fb-4cb1-9109-93f99d2a0482",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fef2342f-e315-41c3-afb3-f7029dc9b2d3",
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   "source": [
    "Mamy\n",
    "$$\n",
    "\\frac{\\partial f}{\\partial x}(x,y) = 2x e^{x^2 + y}\n",
    "$$\n",
    "oraz\n",
    "$$\n",
    "\\frac{\\partial f}{\\partial y}(x,y) =e^{x^2 + y}.\n",
    "$$\n",
    "Stąd\n",
    "$$\n",
    "\\frac{\\partial^2 f}{\\partial x^2}(x,y) =  2 e^{x^2 + y} + 4x^2 e^{x^2 + y},\\quad\n",
    "\\frac{\\partial^2 f}{\\partial y^2} (x,y)= e^{x^2 + y}, \\quad\n",
    "\\frac{\\partial^2 f}{\\partial x \\partial y}(x,y) = 2x e^{x^2 + y}\n",
    "\\quad\\text{ oraz }\\quad \\frac{\\partial^2 f}{\\partial y \\partial x}(x,y) = 2x e^{x^2 + y}.\n",
    "$$\n",
    "\n",
    "Zauważamy, że pochodne cząstkowe $\\frac{\\partial^2 f}{\\partial x \\partial y}(x,y)$ oraz   $\\frac{\\partial^2 f}{\\partial y \\partial x}(x,y)$ są sobie równe.\n",
    "To nie jest przypadek: patrz twierdzenie Schwarza na wykładzie."
   ]
  },
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   "cell_type": "markdown",
   "id": "b08b0f21-1684-4afc-9a26-3d72e5d8d7c6",
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   "source": [
    "#### Zadanie 5 \n",
    "Dla funkcji \n",
    "$$f(x,y,z)=x^3y^2z$$\n",
    "oblicz\n",
    "$$\n",
    "\\frac{\\partial^4 f}{\\partial x^4}(x,y,z),\\quad \\frac{\\partial^4 f}{\\partial x^3\\partial z}(x,y,z)\\quad\\text{ oraz }\\quad \\frac{\\partial^4 f}{\\partial z\\partial y \\partial x^2}(x,y,z)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "415fe031-6d7e-413a-b3a1-88f9b970b92b",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a323a470-c82b-4598-bb78-a69d6e9e09b0",
   "metadata": {},
   "source": [
    "Mamy kolejno:\n",
    "\n",
    "$$ \n",
    "\\frac{\\partial f}{\\partial x}(x,y,z)=3x^2y^2z, \\quad \\frac{\\partial^2 f}{\\partial x^2}(x,y,z)=6xy^2z, \\quad \\frac{\\partial^3 f}{\\partial x^3}(x,y,z)=6y^2z.\n",
    "$$\n",
    "Stąd\n",
    "\n",
    "$$\n",
    "\\quad \\frac{\\partial^4 f}{\\partial x^4}(x,y,z)=0.\n",
    "$$\n",
    "Ponadto \n",
    "$$\n",
    "\\frac{\\partial^4 f}{\\partial x^3\\partial z}(x,y,z)=\\frac{\\partial^4 f}{\\partial z\\partial x^3}(x,y,z)=6y^2\n",
    "$$\n",
    "i \n",
    "$$\n",
    "\\frac{\\partial^4 f}{\\partial z\\partial y \\partial x^2}(x,y,z)=12xy.\n",
    "$$"
   ]
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