Gradient And Steepest Descent On The Surface F(x, Y) = 2y + X^3

Introduction

In the realm of multivariable calculus, understanding the behavior of surfaces is crucial. This article delves into the intricacies of the surface defined by the function f(x, y) = 2y + x³ and a specific point P(1, 1, 3) situated on this surface. We will explore the concept of the gradient, a fundamental tool for analyzing the rate and direction of change of a function. Furthermore, we will investigate the path of steepest descent, a trajectory that follows the direction of the most rapid decrease in the function's value. By projecting this path onto the xy-plane, we can gain valuable insights into the surface's characteristics and its behavior in two dimensions. This exploration will not only enhance our understanding of multivariable calculus but also demonstrate its applications in various fields, including optimization, physics, and computer graphics. This analysis provides a comprehensive understanding of the surface's local behavior around point P and the path of steepest descent, highlighting the interplay between gradients, directional derivatives, and curve projections in multivariable calculus.

a. Finding the Gradient of f

To find the gradient of the function f(x, y) = 2y + x³, we need to compute its partial derivatives with respect to x and y. The gradient, denoted as ∇f, is a vector field that points in the direction of the greatest rate of increase of the function. Its components are the partial derivatives of f with respect to each variable. Understanding the gradient is fundamental to navigating the landscape of a multivariable function, as it provides insights into the function's slope and direction of change at any given point. The gradient vector is a cornerstone of multivariable calculus, offering a powerful tool for analyzing and optimizing functions in higher dimensions. Its applications span across various fields, including physics, engineering, and economics, where understanding the rate and direction of change is crucial for decision-making and problem-solving. In the context of optimization, the gradient helps identify the direction in which to adjust parameters to maximize or minimize a function's value. In physics, it is used to determine the direction of forces and fields, while in economics, it aids in understanding the sensitivity of economic indicators to changes in underlying variables.

First, let's calculate the partial derivative of f with respect to x, denoted as ∂f/∂x. This involves differentiating the function f(x, y) = 2y + x³ with respect to x, treating y as a constant. Applying the power rule of differentiation, we get:

∂f/∂x = ∂(2y + x³)/∂x = 0 + 3x² = 3x²

Next, we calculate the partial derivative of f with respect to y, denoted as ∂f/∂y. This involves differentiating f(x, y) = 2y + x³ with respect to y, treating x as a constant. The derivative of 2y with respect to y is 2, and the derivative of x³ with respect to y is 0, since x³ is considered a constant in this context. Therefore:

∂f/∂y = ∂(2y + x³)/∂y = 2 + 0 = 2

Now that we have both partial derivatives, we can express the gradient of f as a vector field:

∇f(x, y) = (∂f/∂x, ∂f/∂y) = (3x², 2)

The gradient ∇f(x, y) = (3x², 2) represents the direction of the steepest ascent of the function f at any point (x, y). The x-component, 3x², indicates how the function changes with respect to x, while the y-component, 2, indicates how the function changes with respect to y. This vector field provides a comprehensive map of the function's local behavior, allowing us to visualize its slope and direction of change across the entire domain. The gradient is a fundamental concept in optimization algorithms, where it is used to iteratively find the maximum or minimum of a function. By following the direction of the gradient, algorithms can efficiently converge to the optimal solution. Furthermore, the gradient plays a crucial role in understanding the geometry of surfaces and level curves, providing insights into their shape and orientation. Its applications extend beyond mathematics and into various scientific and engineering disciplines, making it an indispensable tool for analyzing and manipulating multivariable functions.

b. Finding the Equation of C: Projection of Steepest Descent

To find the equation of C, the projection of the path of steepest descent C' onto the xy-plane, we first need to understand the concept of steepest descent. The path of steepest descent on a surface begins at a point and follows the direction of the negative gradient. This path represents the trajectory along which the function decreases most rapidly. Projecting this path onto the xy-plane gives us a two-dimensional curve that captures the essence of the function's behavior in terms of x and y. The equation of this curve provides valuable information about the surface's topography and its behavior in the horizontal plane. Analyzing the projection of the steepest descent path is crucial for understanding how the surface descends from a given point and how it relates to the underlying xy-plane. This analysis has applications in various fields, including terrain modeling, optimization problems, and fluid dynamics, where understanding the flow of a system along a surface is essential. By determining the equation of the projection, we can gain insights into the surface's geometry and its behavior in two dimensions, providing a powerful tool for visualization and analysis.

At the point P(1, 1, 3), the gradient is:

∇f(1, 1) = (3(1)², 2) = (3, 2)

The path of steepest descent, C', begins at P and moves in the direction opposite to the gradient. Therefore, the direction vector for the path of steepest descent at P is -∇f(1, 1) = (-3, -2). This direction vector indicates the instantaneous direction of the most rapid decrease in the function's value at point P. As we move along the path of steepest descent, the direction will change depending on the gradient at each point. Understanding the negative gradient is crucial for optimization problems, where the goal is to minimize a function's value. By following the direction of the negative gradient, we can iteratively approach the minimum point of the function. The concept of steepest descent is widely used in machine learning algorithms, where it is used to train models by minimizing a loss function. Furthermore, the path of steepest descent provides valuable insights into the geometry of the surface, revealing the trajectory along which the function decreases most rapidly. Analyzing this path allows us to understand the surface's topography and its behavior in the neighborhood of a given point.

Let C be parameterized by (x(t), y(t)). The tangent vector to C must be parallel to the negative gradient of f. Thus, we have the following differential equations:

(x'(t), y'(t)) = -k(3x²(t), 2)

where k is a positive constant. We can choose k = 1 for simplicity, giving us:

x'(t) = -3x²(t)

y'(t) = -2

These differential equations describe the evolution of the x and y coordinates along the path of steepest descent. The equation x'(t) = -3x²(t) indicates that the rate of change of x with respect to t is proportional to the square of x, but in the opposite direction. This reflects the fact that the steeper the slope in the x-direction, the faster x will decrease along the path of steepest descent. Similarly, the equation y'(t) = -2 indicates that y decreases linearly with respect to t. The constant rate of decrease in y reflects the fact that the gradient in the y-direction is constant. Solving these differential equations will give us the parametric equations for the path of steepest descent. These equations will allow us to trace the trajectory of the path and understand how it evolves over time. The solutions to these equations will provide a detailed description of the path of steepest descent and its relationship to the gradient of the function.

Solving the second equation, y'(t) = -2, we get:

y(t) = -2t + C₁

Since the path starts at P(1, 1, 3), we have y(0) = 1. Thus,

1 = -2(0) + C₁ => C₁ = 1

So, y(t) = -2t + 1.

This equation describes the evolution of the y-coordinate along the path of steepest descent. It shows that y decreases linearly with respect to t, starting from an initial value of 1. The constant rate of decrease, -2, reflects the constant gradient in the y-direction. This equation is a key component in understanding the overall trajectory of the path of steepest descent. By combining it with the solution for x(t), we can fully characterize the path and visualize its behavior in the xy-plane. The linear relationship between y and t simplifies the analysis and provides a clear picture of how the path descends in the y-direction. This equation, along with the solution for x(t), allows us to trace the path and understand its relationship to the gradient of the function.

Now, solving the first equation, x'(t) = -3x²(t), we have:

dx/dt = -3x²

Separating variables, we get:

-dx/(3x²) = dt

Integrating both sides:

∫ -dx/(3x²) = ∫ dt

1/(3x) = t + C₂

Since x(0) = 1, we have:

1/(3(1)) = 0 + C₂ => C₂ = 1/3

So, 1/(3x) = t + 1/3. Solving for x:

1/x = 3t + 1

x(t) = 1/(3t + 1)

This equation describes the evolution of the x-coordinate along the path of steepest descent. It shows that x decreases non-linearly with respect to t, starting from an initial value of 1. The equation reflects the fact that the rate of decrease in x depends on the current value of x, as indicated by the original differential equation x'(t) = -3x²(t). This non-linear relationship adds complexity to the path's trajectory, making it curve more rapidly as x decreases. By combining this equation with the solution for y(t), we can fully characterize the path of steepest descent and visualize its behavior in the xy-plane. This equation is crucial for understanding how the path evolves in the x-direction and how it interacts with the changing gradient of the function.

Now we have the parametric equations for the projection C:

x(t) = 1/(3t + 1)

y(t) = -2t + 1

To find the equation of C in the form y = g(x), we can eliminate the parameter t. From the equation for x(t), we can solve for t:

x = 1/(3t + 1)

3t + 1 = 1/x

3t = 1/x - 1

t = (1/x - 1)/3

Substituting this expression for t into the equation for y(t):

y = -2((1/x - 1)/3) + 1

y = (-2/(3x)) + 2/3 + 1

y = -2/(3x) + 5/3

Thus, the equation of C, the projection of the path of steepest descent on the xy-plane, is:

y = -2/(3x) + 5/3

This equation represents a hyperbola in the xy-plane. It describes the path along which the surface f(x, y) = 2y + x³ descends most rapidly, as viewed from above. The equation provides a clear picture of the relationship between x and y along this path. As x approaches zero, y tends to infinity, reflecting the steepness of the descent in that region. This equation is a valuable tool for visualizing and analyzing the behavior of the surface and its path of steepest descent. It provides a concise representation of the path's trajectory and its relationship to the function's gradient. By understanding this equation, we can gain insights into the surface's geometry and its properties in the xy-plane.

Conclusion

In conclusion, we have successfully determined the gradient of the function f(x, y) = 2y + x³ and found the equation of the projection of the path of steepest descent onto the xy-plane. The gradient, ∇f(x, y) = (3x², 2), provides a vector field that indicates the direction of the steepest ascent of the function at any given point. By following the negative gradient, we can trace the path of steepest descent. The projection of this path onto the xy-plane, represented by the equation y = -2/(3x) + 5/3, gives us a hyperbola that captures the essence of the function's behavior in two dimensions. This analysis demonstrates the power of multivariable calculus in understanding the behavior of surfaces and their properties. The concepts of gradients, directional derivatives, and projections are fundamental tools for analyzing and optimizing functions in higher dimensions. By applying these tools, we can gain valuable insights into the geometry of surfaces and their behavior in various contexts. This understanding has applications in a wide range of fields, including optimization, physics, and computer graphics, where the analysis of surfaces and their properties is crucial for problem-solving and decision-making.