- #1
Niles
- 1,866
- 0
Homework Statement
The gradient of f(x,y) = x^2-x+y is:
gradient_f(x,y) = (2x-1 ; 1). To find gradient_f(x,y), I set 2x-1 = 0 and 1 = 0 - so there are no points, where gradient_f(x,y) is zero because of 1 != 0?
The gradient of a function with multiple variables is a mathematical concept that represents the rate of change or slope of the function at a specific point in multiple dimensions. It is a vector that contains the partial derivatives of the function with respect to each variable.
To calculate the gradient of a function with multiple variables, we take the partial derivative of the function with respect to each variable and combine them into a vector. The resulting vector represents the direction and magnitude of the steepest ascent of the function at that point.
The gradient is a crucial concept in multivariable calculus as it allows us to determine the direction of the steepest ascent or descent of a function at a specific point. It also helps us understand the rate of change of the function in different directions, which is essential in optimization and gradient-based algorithms.
Yes, the gradient of a function with multiple variables can be negative. The sign of the gradient depends on the direction of the steepest ascent or descent of the function at a specific point. A negative gradient indicates a decreasing function, while a positive gradient indicates an increasing function.
The gradient is used in various real-life applications, such as machine learning, computer graphics, and physics. In machine learning, it is used to optimize the parameters of a model to minimize the error. In computer graphics, it is used to create smooth and realistic surfaces. In physics, it is used to calculate the force and potential energy of a system.