MAC2313: Calculus 3 Formula Sheet (Exam 2)
Differentiation of Functions of Several Variables
Limits and Continuity Directional Derivatives Gradient
• Two-Path Test: If f (x, y) → L1 along path C1 and • Gradient Vector (∇f ):
f (x, y) → L2 along path C2 as (x, y) → (a, b), where
L1 ̸= L2 , then lim(x,y)→(a,b) f (x, y) does not exist. ∇f (x, y) = ⟨fx , fy ⟩ = fx i + fy j
• Continuity: f is continuous at (a, b) if: ∇f (x, y, z) = ⟨fx , fy , fz ⟩
lim f (x, y) = f (a, b)
(x,y)→(a,b) • Directional Derivative (Du f ): Given a unit vector
u = ⟨a, b⟩:
Du f = ∇f · u = |∇f | cos θ
Partial Derivatives
Note: Ensure u is a unit vector: u = v/∥v∥.
• Notation: fx = ∂f
∂x , fy = ∂f
∂y
• Properties of Gradient:
• Clairaut’s Theorem: If fxy and fyx are continuous on a
disk containing (a, b), then: – Max rate of increase: ∥∇f ∥ (direction of ∇f ).
fxy (a, b) = fyx (a, b) – Max rate of decrease: −∥∇f ∥ (direction of −∇f ).
– Zero change: Direction orthogonal to ∇f .
• Implicit Differentiation: For F (x, y, z) = 0, define z as
a function of x, y: – ∇f is normal (orthogonal) to level curves/surfaces.
∂z Fx ∂z Fy
=− , =−
∂x Fz ∂y Fz Maximum Minimum Values
• Critical Points: (a, b) where ∇f (a, b) = 0 (i.e., fx = 0
Tangent Planes Linear Approxima- and fy = 0) or where partials do not exist.
tion
• Second Derivative Test:
• Plane Equation: For surface z = f (x, y) at (x0 , y0 , z0 ):
D = D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)]2
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
– If D > 0 and fxx > 0 =⇒ Local Minimum.
• Linearization: L(x, y) ≈ f (x, y) near (a, b):
– If D > 0 and fxx < 0 =⇒ Local Maximum.
L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) – If D < 0 =⇒ Saddle Point.
• Total Differential: dz = fx dx + fy dy – If D = 0 =⇒ Test is inconclusive.
• Tangent Plane to Level Surface: For F (x, y, z) = k at • Absolute Extrema on Closed Sets: 1. Find values at
P (x0 , y0 , z0 ): critical points inside the region.
2. Find extreme values on the boundary.
Fx (P )(x − x0 ) + Fy (P )(y − y0 ) + Fz (P )(z − z0 ) = 0
3. The largest is the Max, smallest is the Min.
• Normal Line:
x − x0 y − y0 z − z0 Lagrange Multipliers
= =
Fx (P ) Fy (P ) Fz (P )
• Optimization with Constraint: To max/min f (x, y, z)
The Chain Rule subject to g(x, y, z) = k:
• Case 1 (1 Independent Variable): z = f (x, y), where ∇f = λ∇g and g(x, y, z) = k
x = g(t), y = h(t):
• System of Equations:
dz ∂f dx ∂f dy
= +
dt ∂x dt ∂y dt fx = λgx , fy = λgy , fz = λgz , g=k
• Case 2 (2 Independent Variables): z = f (x, y), where
x = g(s, t), y = h(s, t): • Two Constraints: Maximize f subject to g = k and
h = c:
∂z ∂z ∂x ∂z ∂y ∇f = λ∇g + µ∇h
= +
∂s ∂x ∂s ∂y ∂s
∂z ∂z ∂x ∂z ∂y
= +
∂t ∂x ∂t ∂y ∂t
Differentiation of Functions of Several Variables
Limits and Continuity Directional Derivatives Gradient
• Two-Path Test: If f (x, y) → L1 along path C1 and • Gradient Vector (∇f ):
f (x, y) → L2 along path C2 as (x, y) → (a, b), where
L1 ̸= L2 , then lim(x,y)→(a,b) f (x, y) does not exist. ∇f (x, y) = ⟨fx , fy ⟩ = fx i + fy j
• Continuity: f is continuous at (a, b) if: ∇f (x, y, z) = ⟨fx , fy , fz ⟩
lim f (x, y) = f (a, b)
(x,y)→(a,b) • Directional Derivative (Du f ): Given a unit vector
u = ⟨a, b⟩:
Du f = ∇f · u = |∇f | cos θ
Partial Derivatives
Note: Ensure u is a unit vector: u = v/∥v∥.
• Notation: fx = ∂f
∂x , fy = ∂f
∂y
• Properties of Gradient:
• Clairaut’s Theorem: If fxy and fyx are continuous on a
disk containing (a, b), then: – Max rate of increase: ∥∇f ∥ (direction of ∇f ).
fxy (a, b) = fyx (a, b) – Max rate of decrease: −∥∇f ∥ (direction of −∇f ).
– Zero change: Direction orthogonal to ∇f .
• Implicit Differentiation: For F (x, y, z) = 0, define z as
a function of x, y: – ∇f is normal (orthogonal) to level curves/surfaces.
∂z Fx ∂z Fy
=− , =−
∂x Fz ∂y Fz Maximum Minimum Values
• Critical Points: (a, b) where ∇f (a, b) = 0 (i.e., fx = 0
Tangent Planes Linear Approxima- and fy = 0) or where partials do not exist.
tion
• Second Derivative Test:
• Plane Equation: For surface z = f (x, y) at (x0 , y0 , z0 ):
D = D(a, b) = fxx (a, b)fyy (a, b) − [fxy (a, b)]2
z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )
– If D > 0 and fxx > 0 =⇒ Local Minimum.
• Linearization: L(x, y) ≈ f (x, y) near (a, b):
– If D > 0 and fxx < 0 =⇒ Local Maximum.
L(x, y) = f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) – If D < 0 =⇒ Saddle Point.
• Total Differential: dz = fx dx + fy dy – If D = 0 =⇒ Test is inconclusive.
• Tangent Plane to Level Surface: For F (x, y, z) = k at • Absolute Extrema on Closed Sets: 1. Find values at
P (x0 , y0 , z0 ): critical points inside the region.
2. Find extreme values on the boundary.
Fx (P )(x − x0 ) + Fy (P )(y − y0 ) + Fz (P )(z − z0 ) = 0
3. The largest is the Max, smallest is the Min.
• Normal Line:
x − x0 y − y0 z − z0 Lagrange Multipliers
= =
Fx (P ) Fy (P ) Fz (P )
• Optimization with Constraint: To max/min f (x, y, z)
The Chain Rule subject to g(x, y, z) = k:
• Case 1 (1 Independent Variable): z = f (x, y), where ∇f = λ∇g and g(x, y, z) = k
x = g(t), y = h(t):
• System of Equations:
dz ∂f dx ∂f dy
= +
dt ∂x dt ∂y dt fx = λgx , fy = λgy , fz = λgz , g=k
• Case 2 (2 Independent Variables): z = f (x, y), where
x = g(s, t), y = h(s, t): • Two Constraints: Maximize f subject to g = k and
h = c:
∂z ∂z ∂x ∂z ∂y ∇f = λ∇g + µ∇h
= +
∂s ∂x ∂s ∂y ∂s
∂z ∂z ∂x ∂z ∂y
= +
∂t ∂x ∂t ∂y ∂t
