diff --git a/math/note05.ipynb b/math/note05.ipynb
index fa2a19f0..e4a182bd 100644
--- a/math/note05.ipynb
+++ b/math/note05.ipynb
@@ -33,7 +33,7 @@
     "\n",
     "|                 Dirac notation for vectors                   |                 Dirac notation for functions                 |\n",
     "| :----------------------------------------------------------: | :----------------------------------------------------------: |\n",
-    "| **Ket** $\\mid a \\rangle =(a_1,a_2,..)\\\\ $ **Bra** $\\bra{a} = \\begin{pmatrix}a_1 \\\\ a_2 \\\\  ...\\\\ \\end{pmatrix}$  | **Ket** $\\mid \\psi\\rangle=\\psi(x)\\\\$ $\\\\$ **Bra** $\\langle \\psi \\mid=\\psi(x)^*$  |\n",
+    "| **Ket** $\\mid a \\rangle =(a_1,a_2,..) \\\\ $ **Bra** $\\bra{a} = \\begin{pmatrix}a_1 \\\\ a_2 \\\\  ...\\\\ \\end{pmatrix}$  | **Ket** $\\mid \\psi\\rangle=\\psi(x)\\\\$ $\\\\$ **Bra** $\\langle \\psi \\mid=\\psi(x)^*$  |\n",
     "|  **Example** $\\\\ \\mid a \\rangle =(1, 2i)$ $\\\\\\bra{a} =\\begin{pmatrix} 1 \\\\ -2i \\\\ \\end{pmatrix}$ | **Example**  $\\\\ \\mid \\psi \\rangle=e^{ix^2}$  $\\\\ \\langle \\psi \\mid = e^{-ix^2}$ |"
    ]
   },
@@ -73,7 +73,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Numerical example of Applying the Derivative Matrix for $\\psi(x) = x^2$\n",
+    "#### Numerical example of Applying the Derivative Matrix for $\\psi(x) = x^2$\n",
     "\n",
     "Let’s consider the function $\\psi(x) = x^2$ sampled at 5 points $x = (0, 1, 2, 3, 4)$, giving us:\n",
     "\n",
@@ -161,12 +161,25 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Centered Difference approximation\n",
+    "#### Centered Difference approximation\n",
     "\n",
-    "- A much better approximation is obtained by using centered difference. \n",
+    "- A much better approximation is obtained by using centered difference (difference between one point to the left and one point to the right from a give point) \n",
     "- And taking care of boundary points via higher order evaluation of derivatives. \n",
     "\n",
-    "$$ \\frac{d\\psi}{dx} \\approx \\frac{\\psi(x_{i+1}) - \\psi(x_{i-1})}{2h} $$"
+    "$$ \\frac{d\\psi}{dx} \\approx \\frac{\\psi(x_{i+1}) - \\psi(x_{i-1})}{2h} $$\n",
+    "\n",
+    "This operation can be represented by a matrix. For $n = 5$ grid points, with improved boundary handling, the first derivative matrix $D_1$ using the centered difference method looks like:\n",
+    "\n",
+    "$$\n",
+    "D_1 = \\frac{1}{2h}\n",
+    "\\begin{pmatrix}\n",
+    "-3 & 4 & -1 & 0 & 0 \\\\\n",
+    "-1 & 0 & 1 & 0 & 0 \\\\\n",
+    "0 & -1 & 0 & 1 & 0 \\\\\n",
+    "0 & 0 & -1 & 0 & 1 \\\\\n",
+    "0 & 0 & 1 & -4 & 3\n",
+    "\\end{pmatrix}\n",
+    "$$\n"
    ]
   },
   {
@@ -207,59 +220,36 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "#### Second Derivative as a Matrix\n",
+    "#### Second Derivative as a Matrix (Centered Difference)\n",
     "\n",
-    "Similarly, the second derivative can be approximated by a central difference formula:\n",
+    "Similarly, the second derivative can be approximated by a **centered difference formula** for interior points:\n",
     "\n",
     "$$ \\frac{d^2\\psi}{dx^2} \\approx \\frac{\\psi(x_{i+1}) - 2\\psi(x_i) + \\psi(x_{i-1})}{h^2} $$\n",
     "\n",
-    "The corresponding second derivative matrix $D_2$ for $n = 5$ grid points is:\n",
+    "At the **boundaries**, we use one-sided difference formulas to maintain accuracy:\n",
+    "\n",
+    "- At the first point $x_1$, we approximate using:\n",
+    "\n",
+    "  $$ \\frac{d^2\\psi}{dx^2} \\bigg|_{x_1} \\approx \\frac{\\psi(x_1) - 2\\psi(x_2) + \\psi(x_3)}{h^2} $$\n",
+    "\n",
+    "- At the last point $x_n$, we approximate using:\n",
+    "\n",
+    "  $$ \\frac{d^2\\psi}{dx^2} \\bigg|_{x_n} \\approx \\frac{\\psi(x_{n-2}) - 2\\psi(x_{n-1}) + \\psi(x_n)}{h^2} $$\n",
+    "\n",
+    "The corresponding second derivative matrix $D_2$ for $n = 5$ grid points, with this improved boundary handling, is:\n",
     "\n",
     "$$\n",
     "D_2 = \\frac{1}{h^2}\n",
     "\\begin{pmatrix}\n",
-    "-2 & 1 & 0 & 0 & 0 \\\\\n",
+    "1 & -2 & 1 & 0 & 0 \\\\\n",
     "1 & -2 & 1 & 0 & 0 \\\\\n",
     "0 & 1 & -2 & 1 & 0 \\\\\n",
     "0 & 0 & 1 & -2 & 1 \\\\\n",
-    "0 & 0 & 0 & 1 & -2\n",
+    "0 & 0 & 1 & -2 & 1\n",
     "\\end{pmatrix}\n",
     "$$\n",
     "\n",
-    "Applying this matrix to the vector $\\psi$ gives the second derivative of the function at each point."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "tags": [
-     "hide-input"
-    ]
-   },
-   "outputs": [],
-   "source": [
-    "import numpy as np\n",
-    "\n",
-    "# Define the grid points and function\n",
-    "n = 5\n",
-    "x = np.linspace(0, 4, n)  # Grid points from 0 to 4\n",
-    "h = x[1] - x[0]  # Spacing between points\n",
-    "\n",
-    "psi = x**2  # Function psi(x) = x^2\n",
-    "\n",
-    "# Define the first derivative matrix D_1\n",
-    "D_1 = (1/h) * np.array([\n",
-    "    [-1, 1, 0, 0, 0],\n",
-    "    [0, -1, 1, 0, 0],\n",
-    "    [0, 0, -1, 1, 0],\n",
-    "    [0, 0, 0, -1, 1],\n",
-    "    [0, 0, 0, 0, 0]\n",
-    "])\n",
-    "\n",
-    "print('second derivative matrix \\n', D_1 @ D_1)\n",
-    "\n",
-    "print('second derivative of x^2 \\n', D_1 @ D_1@x)"
+    "Applying this matrix to the vector $\\psi$ gives the second derivative of the function at each point, including the boundary points with improved accuracy."
    ]
   },
   {