Calculus - Multivariable

Multivariable Calculus

Limits and Continuity

For f(x,y) → L as (x,y) → (a,b): $\lim_{(x,y) \to (a,b)} f(x,y) = L$

Test continuity: check if limit equals function value.

Key idea: Limit must be same approaching from any direction.

Partial Derivatives

$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}$

$\frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x,y+h) - f(x,y)}{h}$

Clairaut’s Theorem: ∂²f/∂x∂y = ∂²f/∂y∂x (if both continuous)

Chain Rule

If z = f(x,y) where x = g(t), y = h(t): $\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$

General Chain Rule: For any number of variables.

Directional Derivatives

Directional derivative in direction of unit vector u = (a,b): $D_uf = \frac{\partial f}{\partial x}a + \frac{\partial f}{\partial y}b$

Gradient

Gradient vector: $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$

Properties:

• D_uf = ∇f · u = |∇f|cos θ
• Maximum rate of change = |∇f|
• Direction of ∇f = direction of maximum increase

Tangent Planes

For z = f(x,y), tangent plane at (a,b): $z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$

Linearization

$\Delta z \approx f_x(a,b)\Delta x + f_y(a,b)\Delta y$

Extrema

Critical points: Where f_x = 0 and f_y = 0.

Second Derivative Test: $D = f_{xx}f_{yy} - (f_{xy})^2$

• D > 0, f_{xx} > 0: local minimum
• D > 0, f_{xx} < 0: local maximum
• D < 0: saddle point
• D = 0: inconclusive

Lagrange Multipliers: For constrained optimization: ∇f = λ∇g at optimum.

Multiple Integrals

Double Integral: $\iint_D f(x,y)dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)dy dx$

Volume: $V = \iint_D f(x,y)dA$

Average Value: $f_{avg} = \frac{1}{A(D)}\iint_D f(x,y)dA$

Change of Variables

For transformation T: (u,v) → (x,y)

$\iint_R f(x,y)dA = \iint_S f(x(u,v),y(u,v))\left|\frac{\partial(x,y)}{\partial(u,v)}\right|du dv$

Jacobian: |∂(x,y)/∂(u,v)| = determinant of matrix.

Polar Coordinates

x = r cos θ, y = r sin θ

$\iint_D f(x,y)dA = \int_{\alpha}^{\beta} \int_{h_1(\theta)}^{h_2(\theta)} f(r\cos\theta, r\sin\theta)r \, dr \, d\theta$

Triple Integrals

$V = \iiint_E dV$

Cylindrical: (x=rcosθ, y=rsinθ, z=z) dV = r dr dθ dz

Spherical: (x=ρ sin φ cos θ, y=ρ sin φ sin θ, z=ρ cos φ) dV = ρ² sin φ dρ dφ dθ