Show that the gradient G[f]=∇f defines a linear operator from the space of continuously differen

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Show that the gradient G[f]=∇f defines a linear operator...

Show that the gradient $G[f]=\nabla f$ defines a linear operator from the space of continuously differentiable scalar-valued functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ to the space of continuous vector fields $\mathbf{v}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.

Key Concepts

Linear Operator

A linear operator is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if L is a linear operator, then for any functions f and g, and any scalar c, L[f + g] = L[f] + L[g] and L[c·f] = c·L[f]. In the context of function spaces, establishing that an operator is linear ensures that the structure of the space is maintained under its action.

Gradient Operator

The gradient operator takes a scalar-valued function and returns a vector field composed of its partial derivatives. In two dimensions, it yields a vector of the form (?f/?x, ?f/?y). This operator is inherently linear because differentiation is a linear process, meaning that the derivative of a sum of functions is the sum of their derivatives, and similarly for scalar multiples.

Continuously Differentiable Functions

Continuously differentiable functions are functions that have derivatives which are not only defined but also continuous. This property is essential in ensuring that the gradient produced by the function behaves well (i.e., is continuous) and that the operations of differentiation are valid. It guarantees smooth behavior, which is fundamental in many analytical contexts.

Vector Fields

A vector field in a two-dimensional space assigns a two-dimensional vector to every point in the domain. When a gradient operator is applied to a scalar function, it generates a vector field whose components are the continuous partial derivatives of the function. The continuity of these derivatives ensures that the resulting vector field is continuous, making it suitable for further analytical and geometrical interpretations.

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