You will prove several basic properties of the gradient for...

You will prove several basic properties of the gradient for functions of two variables. In each exercise, assume that $f$ and/or $g$ is differentiable. Prove that $$ \nabla(\alpha f(x, y)+\beta g(x, y))=\alpha \nabla f(x, y)+\beta \nabla g(x, y), $$ where $\alpha$ and $\beta$ are constants.

You will prove several basic properties of the gradient for functions of two variables. In each exercise, assume that $f$ and/or $g$ is differentiable.

Prove that
$$
\nabla(\alpha f(x, y)+\beta g(x, y))=\alpha \nabla f(x, y)+\beta \nabla g(x, y),
$$
where $\alpha$ and $\beta$ are constants.

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Key Concepts

Gradient

The gradient of a function represents the vector of its partial derivatives with respect to its variables. It indicates the direction of steepest ascent in a multi-dimensional space and is fundamental in analyzing the rate and direction of change in functions of several variables.

Differentiability

Differentiability means that the function has well-defined partial derivatives at a point or over a region, which ensures that the gradient exists. It is a necessary condition to apply many results in multivariable calculus, including those involving linearity properties of differentials.

Linearity of Differential Operators

The linearity of an operator such as the gradient means that when it acts on a linear combination of functions, the result is the equivalent linear combination of the gradients of the functions. This property is central to the problem statement and simplifies the analysis of functions composed as linear combinations.

Partial Derivatives

Partial derivatives are the individual derivatives of a function with respect to each variable separately. They constitute the components of the gradient vector and are essential in understanding the behavior of multivariable functions in each independent direction.

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