Question

$$ \underline{V}(x, y)=e^{x}\left[\begin{array}{l} \sin y \\ \cos y \end{array}\right] $$

Short answer

The gradient vector of the vector field \( \underline{V}(x, y)\) is \( \nabla \underline{V}(x, y) = e^{x}\left[\begin{array}{l}\sin y \\ \cos y \end{array}\right]i + e^{x}\left[\begin{array}{l}\cos y \\ -\sin y \end{array}\right]j \)

Step-by-step solution

Compute the Partial Derivative with Respect to 'x'

The gradient with respect to 'x' is calculated by taking the partial derivative of each component of the vector with respect to 'x'. For the given vector field \( \underline{V}(x, y) = e^{x}\left[\begin{array}{l}\sin y \\ \cos y \end{array}\right]\), the resultant vector after calculating the partial derivatives with respect to 'x' is: \( \frac{\partial \underline{V}}{\partial x} = e^{x}\left[\begin{array}{l}\sin y \\ \cos y \end{array}\right]\)

Compute the Partial Derivative with Respect to 'y'

The gradient with respect to 'y' is calculated by taking the partial derivative of each component of the vector with respect to 'y'. For the given vector field, the resultant vector after calculating the partial derivatives with respect to 'y' is: \( \frac{\partial \underline{V}}{\partial y} = e^{x}\left[\begin{array}{l}\cos y \\ -\sin y \end{array}\right]\)

Altogether

The gradient vector of the vector field \( \underline{V}(x, y)\) is therefore: \( \nabla \underline{V}(x, y) = e^{x}\left[\begin{array}{l}\sin y \\ \cos y \end{array}\right]i + e^{x}\left[\begin{array}{l}\cos y \\ -\sin y \end{array}\right]j \)

Key concepts

Partial Derivative

Understanding the concept of a partial derivative is crucial when dealing with functions of multiple variables. A partial derivative measures how a function changes as one of its input variables is varied, with all other input variables held constant.

For a function like our vector field \( \textbf{V}(x, y) \), which has two input variables, the partial derivatives with respect to \( x \) and \( y \) would tell us how the vector field changes in the direction of each variable, respectively.

In the given exercise, computing the partial derivative of each component of \( \textbf{V}(x, y) \) with respect to \( x \) means taking the derivative while treating \( y \) as a constant. Similarly, when computing the partial derivative with respect to \( y \), \( x \) is treated as constant. This step-by-step differentiation gives us insight into the local behavior of our vector field.

Gradient Vector

The gradient vector is a fundamental concept in vector calculus, representing the direction and rate of the steepest ascent of a function at a given point. To put it simply, if you were standing on a hillside represented by a function, the gradient vector would point in the direction of the steepest slope.

Following the exercise, once we've calculated the partial derivatives with respect to \( x \) and \( y \), the next step is to combine these components into one gradient vector. The \( i \) and \( j \) in the solution represent the unit vectors in the direction of the \( x \) and \( y \) axes, respectively. So, the gradient vector, denoted as \( abla \textbf{V}(x, y) \), succinctly captures how the function \( \textbf{V}(x, y) \) changes in both the \( x \) and \( y \) directions.

Vector Field

A vector field is a map that assigns a vector to each point in a subset of space. In physics, vector fields are used to represent various phenomena such as gravitational fields, electric fields, and fluid flow.

Our exercise deals with a vector field \( \textbf{V}(x, y) \) that varies depending on the coordinates \( x \) and \( y \). In this context, each point in the plane associates with a vector given by the expression in the exercise. These vectors can indicate direction and magnitude, reflecting how particles would move if placed within the field.

Understanding vector fields helps in visualizing complex systems and in solving high-dimensional problems in engineering and science. The exercise demonstrates computing the change of such a field in terms of its individual coordinates, a foundational skill in analyzing the dynamics within a field.