Question

Find the gradient vector field for the scalar function. (That is, find the conservative vector field for the potential function.) $$ f(x, y)=5 x^{2}+3 x y+10 y^{2} $$

Short answer

The gradient vector field for the function \( f(x, y)=5 x^{2}+3 x y+10 y^{2} \) is \( \n\nabla f= (10x + 3y) \hat{i} + (3x + 20y) \hat{j}. \n\n\)

Step-by-step solution

Find the partial derivatives

Find the partial derivative of the function \( f(x, y)=5 x^{2}+3 x y+10 y^{2} \) with respect to the variables x and y, respectively. \n\n\[\n\frac{\partial f}{\partial x} = 10x + 3y, \n\]\n\[\n\frac{\partial f}{\partial y} = 3x + 20y.\n\]

Create the gradient vector field

The gradient vector field for a scalar function is created by combining the partial derivatives as follows: \n\n\[ \n\nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j}, \n\n\] where \( \hat{i} \) and \( \hat{j} \) are the unit vectors in the x and y directions respectively. So in this case, \n\n\[ \n\nabla f = (10x + 3y) \hat{i} + (3x + 20y) \hat{j}. \n\n\]

Write out the final gradient vector field

Using the results from Steps 1 and 2, the gradient vector field for the function \( f(x, y)=5 x^{2}+3 x y+10 y^{2} \) is \n\n\[ \n\nabla f = (10x + 3y) \hat{i} + (3x + 20y) \hat{j}. \n\n\]

Key concepts

Scalar Function

In the context of multivariable calculus, a scalar function is a type of function that assigns a real number to every point in a space. For example, the function described in the exercise, f(x, y) = 5x^2 + 3xy + 10y^2, assigns a real number to each point on the xy-plane. These types of functions are fundamental in that they provide the landscape upon which vector fields, like gradients, are formed.

The value of a scalar function at a given point reflects something about that point's 'position' in the function's domain. For instance, when we evaluate f(x, y) at a specific coordinate, we are effectively determining the 'height' of our function's surface at that point. The concept of scalar functions is pivotal because it serves as a starting point for understanding more complex calculus concepts, such as the gradient.

Partial Derivatives

The partial derivatives of a scalar function provide a way to see how the function changes as we move along one of the axes, keeping all other variables constant. Specifically, for our function f(x, y) = 5x^2 + 3xy + 10y^2, we need to understand how f changes in the x-direction, independently of y, and vice versa. This is achieved by taking the derivative of f with respect to x while treating y as a constant, and then doing the same with y while treating x as a constant.

Mathematically, we express this as:

• For the x-direction: \(\frac{\partial f}{\partial x} = 10x + 3y\).
• For the y-direction: \(\frac{\partial f}{\partial y} = 3x + 20y\).

These partial derivatives are crucial as they will eventually compose the components of our gradient vector.

Conservative Vector Field

A conservative vector field is a special type of vector field that is the gradient of some scalar function, like our f(x, y). One of the key characteristics of conservative vector fields is that the line integral from one point to another is independent of the path taken. This fact is analogous to conservative forces in physics, such as gravity, where the work done by the force is path-independent.

To determine if a vector field is conservative, one often checks if the field can be expressed as the gradient of a potential function. In the case of the exercise, the gradient of f, denoted by \( abla f \), represents the conservative vector field derived from the potential function f(x, y). The components of \( abla f \) are precisely the partial derivatives computed earlier, hinting at the inherent connection between conservative vector fields and scalar functions.

Potential Function

The potential function is a scalar function whose gradient yields a conservative vector field. In simpler terms, it is a 'source' from which the vector field is generated. In our exercise, f(x, y) serves as the potential function because the gradient vector field is obtained from it. The term 'potential' refers to the potential energy in physics, where the potential energy function gives rise to the force fields.

A conservative vector field can be described as the slope or rate of change of the potential function at any given point in the field. For the given function f(x, y), the gradient \( abla f \) indicates the direction of the greatest rate of increase of the function from any point, thus showing how it is intimately related to the notion of potential energy in the field.