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Functions of several variables

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Intros
Lessons
  1. Functions of Several Variables Overview:
  2. 2 Variable Functions
    • How to represent them
    • What they look like Visually
  3. Domain & Range
    • Domain →\to looks at xx and yy
    • Range →\to looks at zz
    • Examples of finding Domain & Range
  4. Traces
    • Traces →\to intersection of surface and planes
    • What it looks like visually
    • Example of finding the Trace
  5. Level Curves
    • Curves on the xyxy-plane with a specific zz value
    • Examples of finding level curves
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Examples
Lessons
  1. Finding Domain & Range
    Find the domain and range of:

    f(x,y)=1x2+y2−4 f(x,y) = \frac{1}{\sqrt{x^2 + y^2 - 4}}

    1. Find the domain and range of:

      f(x,y)=xx2+y2e1−y2f(x,y) = \sqrt{x} \sqrt{x^2+y^2} e^{1-y^2}

      1. Find the domain and range of:

        f(x,y)=ln(x2−1)f(x,y) = ln(x^2 - 1)

        1. Finding the Traces
          Find and graph the trace of the xzxz plane of

          f(x,y)=xx+y2f(x,y) = \sqrt{x}\sqrt{x+y^2}

          1. Find and graph the trace of the yzyz plane of

            5x+2y+z=35x + 2y + z = 3

            1. Find the level curve of f(x,y)=x2+y2f(x,y) = \sqrt{x^2 + y^2} at z0=1z_0 = 1 and z1=2z_1 = 2
              1. Find and graph the level curve of 2z+4y2−x=02z + 4y^2 - x = 0 at z0=0z_0 = 0.
                Topic Notes
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                Notes:

                2 Variable Functions
                Functions with two variables are written in 2 forms:

                z=f(x,y)z = f(x,y)
                F(x,y,z)=0F(x,y,z) = 0


                Domain & Range
                The domain and range of a function f(x,y)f(x,y) is as follows:

                D={(x,y)∈R2:f(x,y)} D = \{ (x,y) \in \mathbb{R}^2 : f(x,y) \}
                R={z∈R:f(x,y)=z}R = \{ z \in \mathbb{R} : f(x,y) = z \}


                Trace

                A trace of a surface is a set of intersecting points which is created by a surface intersecting one of the following planes: xyxy-plane, xzxz-plane, yzyz-plane.

                Trace of the xyxy-plane happens at z=0z=0.
                Trace of the xzxz-plane happens at y=0y=0.
                Trace of the yzyz-plane happens at x=0x=0.

                Level Curve
                A level curve is a curve that is on the xyxy-plane with a specific value z=z0z=z_0.