91Ó°ÊÓ

A scalar function is a mathematical expression that assigns a single value to every point in space described by coordinates like \( x, y, z \). In this context, the goal is to find a scalar function \( f(x, y, z) \) such that its gradient equals the given vector field \( \mathbf{F} \).
Scalar functions differ from vector functions in that they result in a single numerical output rather than a vector. They are easy to visualize as surfaces in three-dimensional space.
For example, when we found the potential function \( f(x, y, z) = e^{y+2z}x + C \), the result is a scalar for each set of inputs \( x, y, z \). Each unique combination provides a specific value which can be understood as height on a continuous surface if plotted graphically. This concept becomes essential in physics, often representing potential energy or electric potential in spatial fields.

Vector fields are mathematical constructs where a vector is assigned to every point in space. They are represented as functions that produce a vector output instead of a scalar one.
In our exercise, we are dealing with the vector field \( \mathbf{F} = e^{y+2z}(\mathbf{i} + x\mathbf{j} + 2x\mathbf{k}) \). This field assigns a vector containing three components at every coordinate \( (x, y, z) \).

• The \( \mathbf{i} \)-component is \( e^{y+2z} \).
• The \( \mathbf{j} \)-component is \( xe^{y+2z} \).
• The \( \mathbf{k} \)-component is \( 2xe^{y+2z} \).

The task in this problem is to determine which scalar function produces this vector field when the gradient operator is applied, showcasing how vector fields can be understood as derivatives of potential functions. They are instrumental in modeling physical phenomena like magnetic fields, wind patterns, or fluid flow.

Partial derivatives are derivatives of multivariable functions, taken with respect to one variable at a time while keeping others constant. They are crucial in finding the gradient of a scalar field.
In our scenario, the partial derivatives must match the respective components of the vector field \( \mathbf{F} \).

• \( \frac{\partial f}{\partial x} = e^{y+2z} \): We find how \( f \) changes with small changes in \( x \) while \( y \) and \( z \) are constant.
• \( \frac{\partial f}{\partial y} = xe^{y+2z} \): This shows \( f \)'s responsiveness to changes in \( y \).
• \( \frac{\partial f}{\partial z} = 2xe^{y+2z} \): We learn how \( f \) varies with \( z \).

By integrating these derivatives step-by-step, one reconstitutes the original scalar function \( f \). This process underscores the foundational principle of how multivariable calculus allows us to peel back the layers of complex functions, revealing hidden structures and simple forms.