Вивчайте The Gradient Vector

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Свайпніть щоб показати меню

The gradient vector is a fundamental concept in multivariate calculus, representing the direction and rate of the steepest ascent of a scalar-valued function of several variables. If you have a function $f(x, y, ..., n)$ , the gradient of $f$ at any point is a vector composed of the partial derivatives of $f$ with respect to each variable. Mathematically, for a function $f(x, y)$ , the gradient is defined as:

∇f(x, y) = \left[\frac{∂f}{∂x}, \frac{∂f}{∂y}\right]

where $\frac{\raisebox{1pt}{$∂f$}}{\raisebox{-1pt}{$∂x$}}$ and $\frac{\raisebox{1pt}{$∂f$}}{\raisebox{-1pt}{$∂y$}}$ are the partial derivatives of $f$ with respect to $x$ and $y$ , respectively. The gradient vector always points in the direction where the function increases most rapidly, and its length tells you how fast the function increases in that direction.

Some important properties of the gradient vector include:

The gradient is perpendicular (normal) to the level curves (contours) of the function;
If the gradient at a point is zero, that point is a critical point (which could be a maximum, minimum, or saddle point);
The gradient can be generalized to functions of more than two variables by including all partial derivatives as components of the vector.

To compute the gradient, you simply calculate all the first-order partial derivatives with respect to each variable at the point of interest.


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import numpy as np
import matplotlib.pyplot as plt

# Define the sample function
def f(x, y):
    return x**2 + y**2

# Compute the gradient of f
def grad_f(x, y):
    df_dx = 2 * x
    df_dy = 2 * y
    return np.array([df_dx, df_dy])

# Create a grid of points
x = np.linspace(-2, 2, 20)
y = np.linspace(-2, 2, 20)
X, Y = np.meshgrid(x, y)

# Compute gradient at each point
U = 2 * X
V = 2 * Y

plt.figure(figsize=(6, 6))
plt.quiver(X, Y, U, V, color="blue", angles='xy')
plt.title("Gradient Vector Field for f(x, y) = x² + y²")
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.show()

Завдання

Проведіть, щоб почати кодувати

Calculate the gradient vector of the function $f(x, y) = x^3 + y^2$ at a given point using the sympy library in the global scope.

Define the symbols x and y utilizing sympy.symbols().
Define the mathematical expression $f = x^3 + y^2$ .
Compute the partial derivative of $f$ with respect to x and y utilizing sympy.diff().
Evaluate both derivatives at the coordinates provided by the val_x and val_y variables utilizing the .subs() method.
Assign the evaluated results to the variables grad_x and grad_y as floats.
Print the result as a tuple (grad_x, grad_y).

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