Unit 5: Multivariate Calculus

I. Orientation — Functions, surfaces, and local change

Multivariate calculus extends single-variable calculus to quantities depending on two or more independent variables. For a function (z=f(x,y)), the pair ((x,y)) identifies a point in the domain and (z) gives its height above the (xy)-plane. Limits describe nearby behavior, derivatives describe local change, and optimization identifies extreme values.

• Independent variables: (x) and (y) may vary separately unless a constraint is imposed.
• Dependent variable: (z=f(x,y)) changes in response to changes in (x) and (y).
• Neighborhood: Approaching ((a,b)) means approaching along every possible path in the (xy)-plane.
• Partial derivatives: (f_x) measures change in the (x)-direction with (y) fixed; (f_y) measures change in the (y)-direction with (x) fixed.
• Differentiability: A differentiable function has a reliable linear approximation near a point.
• Regularity assumption: Continuity of required derivatives is commonly used to justify interchange of mixed partial derivatives and application of optimization tests.

II. Limit, continuity and differentiability of functions of two variables — Local behavior

A. Limit, continuity and differentiability of functions of two variables

These concepts determine whether (f(x,y)) behaves predictably near a point ((a,b)).

• Limit: The statement
  
  TEXT

    lim_(x,y)->(a,b) f(x,y) = L

  
  means that (f(x,y)) approaches (L) along every path as ((x,y)) approaches ((a,b)), excluding the point itself.
• Path test: If two paths give different values, the limit does not exist. For example, for
  
  TEXT

    f(x,y) = xy/(x^2+y^2)

  
  the paths (y=x) and (y=-x) produce (1/2) and (-1/2), so the limit at ((0,0)) fails.
• Continuity: (f) is continuous at ((a,b)) when
  
  TEXT

    lim_(x,y)->(a,b) f(x,y) = f(a,b).

  
  Polynomial functions are continuous everywhere; rational functions are continuous where their denominators are nonzero.
• Partial derivatives:
  
  TEXT

    f_x = ∂f/∂x,    f_y = ∂f/∂y

  
  are calculated by treating the other variable as constant. For (f=x^2y+3y), (f_x=2xy) and (f_y=x^2+3).
• Differentiability: Near ((a,b)), differentiability requires
  
  TEXT

    f(a+Δx,b+Δy) - f(a,b)
    = f_x(a,b)Δx + f_y(a,b)Δy + ε_1Δx + ε_2Δy,

  
  where (\varepsilon_1,\varepsilon_2\to0) as ((\Delta x,\Delta y)\to(0,0)).
• Sufficient condition: If (f_x) and (f_y) exist in a neighborhood and are continuous at ((a,b)), then (f) is differentiable there. Differentiability implies continuity, but continuity alone does not guarantee differentiability.
• Tangent plane: At ((a,b,f(a,b))), the linear approximation is
  
  TEXT

    z - f(a,b) = f_x(a,b)(x-a) + f_y(a,b)(y-b).

III. Chain rule — Differentiating composite functions

A. Chain rule

The chain rule expresses how a dependent quantity changes when its variables depend on one or more parameters.

• Two-variable, one-parameter form: If (z=f(x,y)), (x=x(t)), and (y=y(t)), then
  
  TEXT

    dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).

• Two independent parameters: If (x=x(u,v)) and (y=y(u,v)), then
  
  TEXT

    ∂z/∂u = f_x x_u + f_y y_u,
    ∂z/∂v = f_x x_v + f_y y_v.

• Meaning: Each term is a partial sensitivity multiplied by the corresponding rate of change. The total change is the sum of these pathways.
• Worked example: For (z=x^2+y^2), (x=t^2), and (y=\sin t),
  
  TEXT

    dz/dt = 2x(2t) + 2y(cos t)
           = 4t^3 + 2 sin(t)cos(t).

• Matrix form: Jacobian matrices organize the same rule as
  
  TEXT

    [z_u] = [f_x  f_y][x_u]
    [z_v]   [f_x  f_y][x_v]

  
  with dimensions adjusted when several outputs are present.

IV. Change of variables — Re-expressing a problem in new coordinates

A. Change of variables

Change of variables replaces ((x,y)) by more convenient coordinates ((u,v)), often simplifying boundaries or integrands in double integrals.

• Transformation: A typical transformation is
  
  TEXT

    x = x(u,v),    y = y(u,v).

  
  The original region (R) in the (xy)-plane corresponds to a region (S) in the (uv)-plane.
• Area element: The double integral transforms according to
  
  TEXT

    ∬_R f(x,y) dx dy
    = ∬_S f(x(u,v),y(u,v)) |∂(x,y)/∂(u,v)| du dv.

• Polar coordinates: For
  
  TEXT

    x = r cos θ,    y = r sin θ,

  
  the area element is (dx\,dy=r\,dr\,d\theta). This is effective for circular regions and expressions involving (x^2+y^2=r^2).
• Boundary advantage: The region (x^2+y^2\le a^2) becomes (0\le r\le a), (0\le\theta\le2\pi), replacing a curved boundary by constant limits.
• Conditions: The transformation should be sufficiently smooth and one-to-one, except possibly on boundary sets of zero area; its Jacobian must not vanish in the interior.
• Orientation: Absolute value is used for area and volume because geometric measure cannot be negative.

V. Euler's theorem for homogeneous equations — Scaling structure

A. Euler's theorem for homogeneous equations

Euler's theorem relates the degree of a homogeneous function to a weighted sum of its partial derivatives.

• Homogeneous function: (f(x,y)) is homogeneous of degree (n) if
  
  TEXT

    f(tx,ty) = t^n f(x,y)

  
  for admissible (t). For (x^2+3xy+y^2), every term has degree (2), so the function has degree (2).
• Theorem: If (f) is differentiable and homogeneous of degree (n), then
  
  TEXT

    x f_x + y f_y = n f.

• Derivation: Differentiate (f(tx,ty)=t^n f(x,y)) with respect to (t), then set (t=1):
  
  TEXT

    x f_x(x,y) + y f_y(x,y) = n f(x,y).

• Worked example: For (f=x^2+3xy+y^2),
  
  TEXT

    f_x=2x+3y,    f_y=3x+2y,
    x f_x+y f_y = 2x^2+6xy+2y^2 = 2f.

• Higher-order form: If (f) has continuous second partial derivatives,
  
  TEXT

    x²f_xx + 2xyf_xy + y²f_yy = n(n-1)f.

  
  This follows by differentiating the first Euler identity and combining the resulting equations.
• Limitation: The theorem applies only when all terms have the same total degree; mixed-degree expressions are not homogeneous.

VI. Jacobians — Local scaling and coordinate dependence

A. Jacobians

A Jacobian is the determinant of first partial derivatives and measures local area scaling, while also providing a compact description of multivariable transformations.

• Definition: For (u=u(x,y)) and (v=v(x,y)),
  
  TEXT

    ∂(u,v)/∂(x,y)
    = | u_x  u_y |
      | v_x  v_y |
    = u_xv_y-u_yv_x.

• Geometric meaning: A small rectangle (dx\,dy) is locally transformed into a parallelogram whose area is approximately
  
  TEXT

    |∂(u,v)/∂(x,y)| dx dy.

• Reciprocal relation: When the transformation is invertible and the determinant is nonzero,
  
  TEXT

    ∂(x,y)/∂(u,v) = 1 / [∂(u,v)/∂(x,y)].

• Polar example: For (x=r\cos\theta), (y=r\sin\theta),
  
  TEXT

    ∂(x,y)/∂(r,θ)
    = | cosθ  -r sinθ |
      | sinθ   r cosθ  |
    = r.

• Singularity: A zero Jacobian indicates local collapse of area and prevents direct use of the ordinary inverse transformation theorem.
• Functional dependence: If (u) and (v) are functionally dependent, their Jacobian is zero; locally, one variable can be expressed as a function of the other.

VII. Extrema of functions of two variables — Optimization without constraints

A. Extrema of functions of two variables

Extrema are located by finding stationary points and classifying the local surface shape.

• Critical point: A point ((a,b)) is critical if
  
  TEXT

    f_x(a,b)=0 and f_y(a,b)=0,

  
  or if one of these partial derivatives does not exist.
• Second-derivative test: Define
  
  TEXT

    D = f_xx(a,b)f_yy(a,b) - [f_xy(a,b)]².

• Classification:
  • If (D>0) and (f_{xx}>0), there is a local minimum.
  • If (D>0) and (f_{xx}<0), there is a local maximum.
  • If (D<0), there is a saddle point.
  • If (D=0), the test is inconclusive.
• Worked example: For (f=x^2+y^2-4x-6y), solving (f_x=2x-4=0), (fy=2y-6=0) gives ((2,3)). Since (f{xx}=2), (f{yy}=2), (f{xy}=0), (D=4>0), and (f_{xx}>0), the point is a local minimum.
• Absolute extrema: On a closed and bounded region, a continuous function attains both an absolute maximum and minimum. Interior critical points and boundary points must both be examined.
• Boundary analysis: A boundary curve can be parameterized, such as (x=\cos t), (y=\sin t), reducing the restriction to a one-variable problem.

VIII. Lagrange's method of undetermined multipliers — Extrema under constraints

A. Lagrange's method of undetermined multipliers

Lagrange’s method finds constrained extrema by requiring the objective’s gradient to be parallel to the constraint’s gradient.

• Constraint form: To optimize (f(x,y)) subject to (g(x,y)=c), solve
  
  TEXT

    ∇f = λ∇g,    g(x,y)=c,

  
  where (\lambda) is the Lagrange multiplier.
• Component equations:
  
  TEXT

    f_x = λg_x,
    f_y = λg_y,
    g(x,y)=c.

• Geometric principle: At a constrained extremum, the level curve (f(x,y)=k) is tangent to the constraint curve (g(x,y)=c); therefore their normal vectors (\nabla f) and (\nabla g) are parallel.
• Worked example: Optimize (f=x+y) subject to (x^2+y^2=1). The equations are
  
  TEXT

    1=2λx,    1=2λy,    x²+y²=1.

  
  Hence (x=y), giving (x=y=\pm1/\sqrt2). The maximum is (\sqrt2) at ((1/\sqrt2,1/\sqrt2)), and the minimum is (-\sqrt2) at ((-1/\sqrt2,-1/\sqrt2)).
• Multiple constraints: For (g(x,y,z)=c_1) and (h(x,y,z)=c_2),
  
  TEXT

    ∇f = λ∇g + μ∇h,

  
  with multipliers (\lambda) and (\mu).
• Limitations: The method requires differentiability and may miss singular constraint points where (\nabla g=0). All candidate points must be compared to determine which gives the absolute constrained extrema.