CALCULUS · MANY VARIABLES

Multivariable calculus

When a quantity depends on several inputs, derivatives become gradients and integrals extend across areas and volumes. The central ideas of single-variable calculus survive — but geometry becomes richer.

∇fgradient is perpendicular to level curves

1. Functions of several variables

A function such as \(f(x,y)\) assigns one output to each allowed point \((x,y)\) in its domain. Its graph \(z=f(x,y)\) is typically a surface in three-dimensional space.

A partial derivative measures change in one coordinate direction while holding the other variables fixed.

\[f_x(a,b)=\lim_{h\to0}\frac{f(a+h,b)-f(a,b)}h,\qquad f_y(a,b)=\lim_{h\to0}\frac{f(a,b+h)-f(a,b)}h.\]

For \(f(x,y)=x^2y+3y^2\),

\[f_x=2xy,\qquad f_y=x^2+6y.\]

2. Gradient and directional derivatives

The gradient collects the first partial derivatives:

\[\nabla f=(f_x,f_y)\quad\text{or in three variables}\quad\nabla f=(f_x,f_y,f_z).\]
Direction

\(\nabla f\) points in the direction of steepest increase.

Magnitude

\(\|\nabla f\|\) is the maximum instantaneous rate of increase.

Level sets

The gradient is perpendicular to a smooth level curve or surface.

For a unit vector \(\mathbf u\), the directional derivative is

\[D_{\mathbf u}f=\nabla f\cdot\mathbf u.\]

3. Tangent planes and linearization

If \(f\) is differentiable near \((a,b)\), then its best local linear approximation is

\[L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).\]

The graph \(z=L(x,y)\) is the tangent plane to \(z=f(x,y)\) at \((a,b,f(a,b))\).

Differentiability is stronger than having partial derivatives. Partial derivatives can exist at a point even when no valid tangent-plane approximation exists there.

4. Local extrema and constrained optimization

At an interior differentiable local extremum,

\[\nabla f=\mathbf0.\]

For two variables, the second-derivative test uses

\[D=f_{xx}f_{yy}-(f_{xy})^2.\]
ConditionClassification
\(D>0, f_{xx}>0\)local minimum
\(D>0, f_{xx}<0\)local maximum
\(D<0\)saddle point
\(D=0\)test inconclusive

Lagrange multipliers

To optimize \(f\) subject to \(g(x,y)=c\), solve

\[\nabla f=\lambda\nabla g,\qquad g(x,y)=c.\]

Geometrically, at a constrained extremum the level curves of \(f\) and the constraint are tangent.

5. Double integrals

A double integral accumulates values across a planar region \(R\):

\[\iint_R f(x,y)\,dA.\]

For a rectangular region \([a,b]\times[c,d]\), a continuous function can be integrated iteratively:

\[\int_a^b\int_c^d f(x,y)\,dy\,dx.\]

If \(f=1\), the double integral gives the area of \(R\). If \(f\ge0\), it can represent volume beneath the surface \(z=f(x,y)\). Fubini's theorem gives conditions under which the order of integration may be exchanged.

Worked example

\[\int_0^1\int_0^2(x+y)\,dy\,dx=\int_0^1(2x+2)dx=3.\]

6. Interactive quadratic gradient tool

Explore \(f(x,y)=Ax^2+Bxy+Cy^2+Dx+Ey+F\) at a point \((x,y)\).

7. Common mistakes

  • Changing every variable when taking a partial derivative instead of holding the others constant.
  • Assuming \(\nabla f=0\) automatically means a maximum or minimum; saddle points are possible.
  • Using a non-unit direction vector directly in a directional derivative when a rate per unit distance is intended.
  • Forgetting that iterated-integral bounds may depend on the outer variable for nonrectangular regions.

8. Practice

  1. For \(f=x^2+xy+y^2\), find \(\nabla f\).
  2. Find the tangent plane to \(z=x^2+y^2\) at \((1,2,5)\).
  3. Classify the critical point of \(f=x^2-y^2\) at the origin.
Solutions

1) \((2x+y,x+2y)\). 2) \(z=5+2(x-1)+4(y-2)\). 3) Saddle point because \(D=(2)(-2)-0=-4<0\).