# 7. Multi-variable Calculus¶

In [1]:
from IPython.core.display import HTML
css_file = 'css/ngcmstyle.css'

Out[1]:
In [1]:
%matplotlib inline

#rcParams['figure.figsize'] = (10,3) #wide graphs by default
import scipy
import numpy as np
import time
from sympy import symbols,diff,pprint,sqrt,exp,sin,cos,log,Matrix,Function,solve

from mpl_toolkits.mplot3d import Axes3D
from IPython.display import clear_output,display,Math
import matplotlib.pylab as plt


## Definition¶

Suppose that $(x_0, y_0)$ is in the domain of $z = f (x, y)$ 1. the partial derivative with respect to $x$ at $(x_0, y_0)$ is the limit

$$\mathbf{\frac{\partial f}{\partial x} (x_0, y_0) = \lim_{h \rightarrow 0} \frac{f (x_0 + h, y_0) - f (x_0, y_0)}{h} }$$

Geometrically, the value of this limit is the slope of the tangent line of $z = f (x, y)$ in the plane $y = y_0$. And this quantity is the rate of change of $f (x, y)$ at $(x_0, y_0)$ along the $x$-direction.

2. the partial derivative with respect to $y$ at $(x_0, y_0)$ is the limit

$$\mathbf{ \frac{\partial f}{\partial y} (x_0, y_0) = \lim_{k \rightarrow 0} \frac{f (x_0, y_0 + k) - f (x_0, y_0)}{k} }$$

Geometrically, the value of this limit is the slope of the tangent line of $z = f (x, y)$ in the plane $x = x_0$. And this quantity is the rate of change of $f (x, y)$ at $(x_0, y_0)$ along the $y$-direction.

Here is a geometric meaning about partial derivative:

<img src="imgs/8/cal7-11.png" width=80% />

In [2]:
import plotly.graph_objs as go

import plotly
from plotly.offline import init_notebook_mode,iplot
init_notebook_mode()