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Calculus III (Multivariable) Unit 3: Functions of several variables

Functions of several variables have a partial derivative for each input. Each one measures change in a single direction while the others are held fixed.

Covers: Limits & continuity, partial derivatives, tangent planes, chain rule

Study guide

Functions of several variables, like z = f(x, y), take multiple inputs. Their graphs are surfaces, and their level curves f(x, y) = k form contour maps. The domain is a region of the plane.

Limits in two variables must give the same value along every path to the point. To show a limit doesn't exist, find two paths (such as along the x-axis and along y = x) that give different values. A function is continuous where its limit equals its value; polynomials and most familiar combinations are continuous on their domains.

Partial derivatives measure the rate of change in one variable while holding the others constant. To find ∂f/∂x, treat y as a constant and differentiate with respect to x. Second partials include f_xx, f_yy and mixed partials f_xy and f_yx, which are equal when they're continuous (Clairaut's theorem).

The tangent plane to z = f(x, y) at (x₀, y₀) is z − z₀ = f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀). It gives the linear approximation of f near that point, and the total differential dz = f_x dx + f_y dy estimates small changes.

The multivariable chain rule follows every path from the output to the variable: if z = f(x, y) with x and y depending on t, then dz/dt = f_x·dx/dt + f_y·dy/dt. Implicit differentiation follows: dy/dx = −F_x/F_y for F(x, y) = 0.

When several equations tie variables together, take the differential of each one (for example d(x²y) = 2xy dx + x² dy), substitute the point, and solve the resulting linear equations for the ratio you need, such as dy/dx or ∂z/∂x. If z = f(x, y) with x = x(s, t) and y = y(s, t), then ∂z/∂s = f_x·∂x/∂s + f_y·∂y/∂s, and similarly for ∂z/∂t. A tree diagram from z down to s and t shows every term.

If F(x, y, z) = 0 defines z implicitly, then ∂z/∂x = −F_x/F_z and ∂z/∂y = −F_y/F_z. In a linear approximation Δf ≈ f_x Δx + f_y Δy, the input whose partial derivative times its possible change is largest in size has the biggest effect on f.

Key ideas

• Level curves f(x, y) = c show a 3D surface as a contour map.
• ∂f/∂x: treat y as a constant and differentiate in x.
• Clairaut's theorem: mixed partials are equal when they're continuous.
• The tangent plane is the 3D version of the tangent line.
• Multivariable chain rule: add a term for every path from the output to the variable.

Formulas and rules

Common mistakes

• Forgetting to hold the other variable constant.
• Showing a limit exists by checking just one path.

Words to know

Partial derivative ∂f/∂x

Differentiate in x, treating other variables as constants.

Clairaut's theorem

Continuous mixed partials are equal: f_xy = f_yx.

Tangent plane to z = f(x, y) at (x₀, y₀)

z − z₀ = f_x(x − x₀) + f_y(y − y₀)

Chain rule for z = f(x(t), y(t))

dz/dt = f_x·x′(t) + f_y·y′(t)

Two-path test

Different limits along two paths → the limit doesn't exist.

Function of two variables

z = f(x, y), graphed as a surface.

Level curve

The curve f(x, y) = k; together they form a contour map.

Limit in two variables

Must give the same value along every path.

Mixed partial derivatives

f_xy and f_yx.

Linear approximation (2 variables)

f(x, y) ≈ f(a, b) + f_x(x − a) + f_y(y − b)

Total differential

dz = f_x dx + f_y dy

Implicit differentiation (multivariable)

dy/dx = −F_x/F_y for F(x, y) = 0.

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.