Partial Derivatives

import sympy as sp
x, y = sp.symbols('x y')

# Define the function
f = x * y**2


# Calculate the partial derivatives
partial_x = sp.diff(f, x)  # Partial derivative with respect to x
partial_y = sp.diff(f, y)  # Partial derivative with respect to y

# Display the results
print("Partial derivatives wrt x and y:")
display(partial_x, partial_y)

Partial derivatives wrt x and y:

\displaystyle y^{2}

\displaystyle 2 x y

import numpy as np
import plotly.graph_objects as go

# Define the range for x and y
pi = np.pi
x = np.linspace(-2 * pi, 2*pi, 100)
y = np.linspace(-2*pi, 2*pi, 100)
x, y = np.meshgrid(x, y)

# Calculate z based on the given function
z =  np.sin(x) + np.sin(y)


color1 = 'orange'
color2 = 'orange'
# Create the surface plot with custom colors
fig = go.Figure(data=[go.Surface(z=z, x=x, y=y, colorscale='viridis', showscale=False, hoverinfo = 'skip')])

# Update layout for background and titles
fig.update_layout(
    title='Surface Plot of z = y^2 cos(xy)',
    scene=dict(xaxis_title='X', yaxis_title='Y', zaxis_title='Z')
)

showbackground = False

fig.update_layout(scene = dict(
                    xaxis = dict(
                         backgroundcolor="rgb(200, 200, 230)",
                         gridcolor="white",
                         showbackground=showbackground,
                         zerolinecolor="black",),
                    yaxis = dict(
                        backgroundcolor="rgb(230, 200,230)",
                        gridcolor="white",
                        showbackground=showbackground,
                        zerolinecolor="black"),
                    zaxis = dict(
                        backgroundcolor="rgb(230, 230,200)",
                        gridcolor="white",
                        showbackground=showbackground,
                        zerolinecolor="black",),),
                    # width=700,
                    margin=dict(
                    r=10, l=10,
                    b=10, t=26)
                  )


# Show the plot
fig.show()

import numpy as np
import plotly.graph_objects as go

# Define the paraboloid z = x^2 + y^2
def paraboloid(x, y):
    return x**2 + y**2

# Create a grid for x and y
x = np.linspace(-2.5, 2.5, 100)
y = np.linspace(-2.5, 2.5, 100)
x, y = np.meshgrid(x, y)
z = paraboloid(x, y)

mask = x**2 + y**2 <= 4
z[~mask] = np.nan  # Set values outside the circle to NaN for visualization


# Define the intersection plane x = 1
x_intersection = 1
y_intersection = np.linspace(-2, 2, 100)
z_intersection = paraboloid(x_intersection, y_intersection)

# Point of tangency on the intersection
y_tangent_point = 0.5  # You can choose another point on the parabola
z_tangent_point = paraboloid(x_intersection, y_tangent_point)

# Calculate the slope of the tangent line
# Derivative of z with respect to y: dz/dy = 2y
tangent_slope = 2 * y_tangent_point  # at y = 0, this is 0

# Create the tangent line in the plane z = k + m(y - y0)
y_tangent = np.linspace(y_tangent_point - 1, y_tangent_point + 1, 100)
z_tangent = z_tangent_point + tangent_slope * (y_tangent - y_tangent_point)

# Create the figure
fig = go.Figure()

# Add paraboloid 
color1 = 'orange'
color2 = 'orange'
fig.add_trace(go.Surface(z=z, x=x, y=y, colorscale=[[0, color1], [1,color2]], opacity=0.7, name='Paraboloid', hoverinfo = 'skip'))

# Add intersecting plane
fig.add_trace(go.Scatter3d(x=[x_intersection]*len(y_intersection), 
                             y=y_intersection, 
                             z=z_intersection, 
                             mode='lines', 
                             name='Intersection Parabola', 
                             line=dict(color='red', width=5)))

# Add tangent line
fig.add_trace(go.Scatter3d(x=[x_intersection]*len(y_tangent), 
                             y=y_tangent, 
                             z=z_tangent, 
                             mode='lines', 
                             name='Tangent Line', 
                             line=dict(color='blue', width=5)))

# Update layout
fig.update_layout(title='Paraboloid with Intersection and Tangent Line',
                  scene=dict(xaxis_title='X', yaxis_title='Y', zaxis_title='Z'),
                  width=800, height=600,
                  paper_bgcolor='white')

# Show the plot
fig.show()