91Ó°ÊÓ

In vector calculus, a gradient field is a special type of vector field. In a gradient field, each vector points in the direction of the steepest increase of a scalar function. This means that if you imagine a hilly landscape where each point has a corresponding height, the vectors in the gradient field show you the steepest direction you should take to go uphill.
To determine if a vector field is a gradient field, we look for a scalar function, usually denoted as \( f(x, y) \), whose gradient (denoted as \( abla f \)) matches the given vector field. The gradient \( abla f \) is calculated using the partial derivatives of \( f \) with respect to each variable. If the vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is a gradient field, then there exists a function \( f \) such that:
\[ abla f = (f_x, f_y) = (P, Q) \]
This implies that:
\[ P = \frac{\partial f}{\partial x} \quad \text{and} \quad Q = \frac{\partial f}{\partial y} \]
Moreover, for \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) to be a gradient field, an important condition must be satisfied:
\[ \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y} \].
If this condition holds, then the vector field is indeed a gradient field.

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of multiple variables. A partial derivative of a function measures how the function changes as one of the input variables changes, while all other input variables are kept constant. For a function \( f(x, y) \), the partial derivatives are denoted as \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).
Here’s a step-by-step guide for finding partial derivatives:

• Identify the function and the variable you are differentiating with respect to.
• Treat all other variables as constants.
• Differentiation will follow the same rules as single-variable calculus.

For example, if \( f(x, y) = 2xy + y^2 + 1 \), the partial derivative with respect to \( y \) is:
\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(2xy + y^2 + 1) = 2x + 2y \].
Similarly, for a function \( g(x, y) = x^2 + 2xy + x \), the partial derivative with respect to \( x \) is:
\[ \frac{\partial g}{\partial x} = \frac{\partial}{\partial x}(x^2 + 2xy + x) = 2x + 2y + 1 \].
These derivatives are used to check the condition for gradient fields, ensuring the slopes in different directions match what is expected in a gradient.

A scalar function assigns a single real number to every point in space. For functions of two variables, like \( f(x, y) \), each input pair \( (x, y) \) gets mapped to a single output value. Scalar functions are critical in vector calculus because they help us describe quantities that have magnitude but no direction at every point in a region.
When dealing with gradient fields, the scalar function is essential because the gradient of this scalar function forms the vector field. If you have a scalar function \( f(x, y) \), its gradient \( abla f \) is a vector field composed of its partial derivatives:
\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \].
In the context of our exercise, the challenge is to determine if a given vector field \( \left(2xy + y^2 + 1\right) \mathbf{i} + \left(x^2 + 2xy + x\right) \mathbf{j} \) is a gradient field of some scalar function. We do this by checking if there exists a scalar function \( f(x, y) \) such that its gradient matches the given vector field. This involves ensuring the partial derivatives \( \frac{\partial Q}{\partial x} \) and \( \frac{\partial P}{\partial y} \) are equal.