91Ó°ÊÓ

In multivariable calculus, differentiability is an essential concept that provides information about the behavior of a function at a given point. A function of several variables, like \( f(x, y) \), is deemed differentiable at a point \((x_0, y_0)\) if it can be approximated very closely by a linear function around that point. This implies that as \((\Delta x, \Delta y)\) tends to zero, \(\Delta f\) should not be erratic. Instead, it should resemble a consistent direction, which can be expressed through partial derivatives.

Differentiability ensures that we can create a tangent plane at \((x_0, y_0)\) and predict how the function will change in small neighborhoods of that point. Hence, it involves both the function itself and its first partial derivatives. A function may be continuous everywhere but not necessarily differentiable. Differentiability requires the existence of well-defined partial derivatives that yield a specific linear approximation.

• For a function \( f(x, y) \) to be differentiable at \((x_0, y_0)\), \(f\) must be locally linear.
• The key idea is the approximation: \(\Delta f \approx f_x(x_0, y_0)\Delta x + f_y(x_0, y_0)\Delta y\).
• Not all continuous functions are differentiable due to unforeseen behaviors like sharp turns or cusps.

Continuity of a multivariable function \(f(x, y)\) at a point \((x_0, y_0)\) is fundamental to ensure that the value of \(f\) does not drastically change. It means that for every small change in the input (\(x, y\)), there is a small change in the output, \(f(x, y)\). This smooth transition is crucial for the discussion of limits and differentiability. If \(f\) is not continuous, the limit \((\Delta x, \Delta y) \rightarrow (0, 0)\) cannot be evaluated easily.

For a function to be continuous at a point, it must satisfy that the limit of \(f(x, y)\) as \((x, y)\) approaches \((x_0, y_0)\) is equal to \(f(x_0, y_0)\). This is often expressed as:

- \(\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)\)

Continuity of a function in every neighborhood is a prior requisite for differentiability, but it alone doesn't guarantee differentiability; the function must also be well-behaved in terms of creating stable derivatives. Many functions in multivariable calculus show continuity without being differentiable at all points.

• Continuity ensures no sudden jumps in function values.
• If continuity fails, any claim on differentiability becomes suspect.
• Maintains smoothness in the transition as points approach \((x_0, y_0)\).

Partial derivatives provide the rate of change of a multivariable function, \(f(x, y)\), with respect to one variable while keeping the other constant. These derivatives are like the slopes of tangent lines touching the surface described by \(f(x, y)\) when observing changes along each axis individually. The partial derivative with respect to \(x\), denoted as \(f_x(x, y)\), shows the function's behavior along the \(x\)-axis, and \(f_y(x, y)\) does the same for the \(y\)-axis.

To compute a partial derivative, effectively treat all other variables as constants and differentiate with respect to the variable of interest. These computations are vital for evaluating and estimating changes around a point on a multivariable function. In the context of differentiability, partial derivatives are crucial because together they construct the gradient vector, a pivotal component in determining linearity.

• The notations \(f_x\) and \(f_y\) represent the rates of change along the \(x\) and \(y\) directions.
• Partial derivatives help form the linear approximation: \(\Delta f \approx f_x \Delta x + f_y \Delta y\).
• Existence of partial derivatives in itself is not sufficient for differentiability; they must combine smoothly to form tangent planes.