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Dated: 30-10-2024

Geometry of Continuous Functions

One Variable

The graph can be imagined as if it was drawn using a pen and you keep moving the pen on the paper without lifting it up.

Two Variables

The graph can be imagined as constructed from a thin sheet of clay which is pinched and hollowed into peaks and valleys.

Continuity of functions1 in 2 Variables

  1. \(f(x_0, y_0)\) is defined
  2. \(\lim_{(x, y) \to (x_0, y_0)} f(x, y)\) exists
  3. \(\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)\)

The first point makes sure that there is no hole in the sheet.

Continuity of functions1 in 3 Variables

  1. \(f(x_0, y_0, z_0)\) is defined
  2. \(\lim_{(x, y, z) \to (x_0, y_0, z_0)} f(x, y)\) exists
  3. \(\lim_{(x, y, z) \to (x_0, y_0, z_0)} f(x, y, z) = f(x_0, y_0, z_0)\)

Rules of continuous functions2

If \(g(x)\) and \(h(y)\) are continuous functions2 in one variable and \(f(x, y)\) depends on \(g(x)\) and \(h(y)\) then

  1. Product
  2. Sum
  3. Difference
  4. Quotient
  5. Composition

of these functions1(\(f(x, y)\)) is also continuous.

Partial Derivatives

Let \(f(x, y)\) be a function.1
If we hold \(y = y_0\) and view \(x\) as the variable then \(f(x, y_0)\) depends on \(x\) alone.
if \(f(x, y_0)\) is differentiable3 at \(x = x_0\) then \(f_x(x_0, y_0)\) is called partial derivative of \(f(x, y)\) with respect to \(x\) at \((x_0, y_0)\).

Similarly, \(f_y(x_0, y_0)\) is partial derivative of \(f(x, y)\) with respect to \(y\) at \((x_0, y_0)\).

Example

Find \(f_x(1, 2)\) and \(f_y(1, 2)\) for

\[f(x,y) = 2x^3y^2 + 2y + 4x\]

Solution

\[f_x(x, y) = 6 x^2 y^2 + 4\]

Reminder: \(y\) is treated like a constant.

\[f_y(x, y) = 4 x^3 y + 2\]

Substituting values, we get

\[f_x(1, 2) = 6(1)^2(2)^2 + 4 = 28\]
\[f_y(1, 2) = 4(1)^3(2) + 2 = 10\]

Example

\[Z = 4x^2 - 2y + 7x^4y^5\]
\[\frac{\partial z}{\partial x} = 8x + 28x^3y^5\]
\[\frac{\partial z}{\partial y} = -2 + 35x^4y^4\]

Example

\[w = x^2 + 3y^2 + 4z^2 - xyz\]
\[\frac{\partial w}{\partial x} = 2x - yz\]
\[\frac{\partial w}{\partial y} = 6y - xz\]
\[\frac{\partial w}{\partial z} = 8z - xy\]

References

Read more about notations and symbols.

  1. Read more about functions. ↩↩↩↩

  2. Read more about continuity. ↩↩

  3. Read more about differentiation. ↩