MAT2615
Calculus in Higher Dimensions
Complete Study Notes & Formula Sheets
Formula sheets · step-by-step methods · worked examples
for every section of Units 1–19
Part 1 – Basic Concepts: Units 1–3
Part 2 – Differentiation: Units 4–11
Part 3 – Integration: Units 12–19
Compiled as a revision companion to the UNISA MAT2615 study guides.
,MAT2615 Study Notes Calculus in Higher Dimensions
Contents
1 Mathematical Preliminaries 2
1.1 1.2–1.3 Sets, Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 1.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Concepts and Operations in n-Dimensional Euclidean Space 4
n
2.1 2.3–2.9 Vector Algebra in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 3
2.2 2.10 The Cross Product in R (only defined in R !) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
n
2.3 2.11 Lines in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
n
2.4 2.12 Planes in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
n
2.5 2.13 Subsets of R : Open, Closed, Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Functions 7
n m
3.1 3.1 Types of R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 3.2 Visualizing Functions: Graphs and Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
n p p m
3.3 3.3 Composition of R → R and R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Limits and Continuity 9
n
4.1 4.2–4.4 Limits of R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 4.5 Limits Along Curves – Proving a Limit Does Not Exist . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 4.6–4.7 Limits of Vector-Valued Functions & Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 The Derivative of a Real-Valued Function of One Variable 11
6 The Derivative of a Vector-Valued Function of One Variable 12
6.1 6.1–6.3 Vector Derivative, Chain Rule, Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . 12
6.2 6.4 Piecewise Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Derivatives of a Real-Valued Function of Several Variables 13
7.1 7.2–7.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.2 7.4–7.5 The Gradient and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3 7.6 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4 7.7 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.5 7.8 Tangent Planes to Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.6 7.9 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.7 7.10 Higher Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8 Derivatives of a Vector Field 17
8.1 8.2 The Derivative Matrix (Jacobian) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.2 8.3–8.4 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3 8.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9 Taylor Polynomials 19
9.1 9.2 Taylor Polynomials for R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2 9.3 Taylor Polynomials for Rn → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10 Optimization 20
2
10.1 10.2 Local Extrema of R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1
,MAT2615 Study Notes Calculus in Higher Dimensions
10.2 10.3 Constrained Optimization – Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11 The Implicit Function Theorem 22
11.1 11.2 IFT for an Equation in 2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.2 11.3 IFT for an Equation in n Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.3 11.4–11.5 The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12 Single Integrals 24
12.1 12.2–12.4 Definite Integrals, FTC, Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13 Double Integrals 25
13.1 13.5 Iterated Integrals over Type 1 and Type 2 Regions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
13.2 13.6 Change of Variables (Jacobian) and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 25
14 Triple Integrals 27
14.1 14.4–14.5 Iterated Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
14.2 14.6 Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
15 Line Integrals of Real-Valued Functions 29
15.1 15.2–15.3 Line Integrals w.r.t. Arc Length, x, and y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
15.2 15.5 Line Integrals over Piecewise Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
16 Line Integrals of Vector Fields 30
16.1 16.2 The Line Integral of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
16.2 16.5–16.7 The Fundamental Theorem of Line Integrals & Path Independence . . . . . . . . . . . . . . 30
17 Surface Integrals 32
17.1 17.2–17.4 Surface Integrals of Scalar Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
18 Flux Integrals 33
18.1 18.2–18.4 Setting Up and Evaluating Flux Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
19 The Theorems of Green, Gauss and Stokes 34
19.1 19.1 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
19.2 19.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
19.3 19.3 Gauss’s Theorem (The Divergence Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2
, MAT2615 Study Notes Calculus in Higher Dimensions
Unit 1 Mathematical Preliminaries
1.2–1.3 Sets, Relations and Functions
x ∈ S: x is an element of S x ∈ / S: x is not an element of S
A ∪ B = {x | x ∈ A or x ∈ B} A ∩ B = {x | x ∈ A and x ∈ B}
B − A = {x ∈ B | x ∈ / A} (complement of A relative to B)
A × B = {(a, b) | a ∈ A, b ∈ B} (product set); R × R = R2 , R × R × R = R3
Function: f : A → B such that every a ∈ A has exactly one image f (a) = b ∈ B.
Domain Df : set of inputs. Image/range f (A) = {f (a) | a ∈ A}.
Onto (surjective): every b ∈ B is hit at least once.
One-to-one (injective): f (x) = f (y) =⇒ x = y.
Invertible ⇐⇒ one-to-one and onto; inverse g satisfies g ◦ f = IA , f ◦ g = IB .
Composition: (f ◦ g)(x) = f (g(x)), domain = all x ∈ Dg with g(x) ∈ Df .
1. Write the relation as pairs (x, y) satisfying some condition.
2. Ask: for a given x, is there exactly one y? If some x gives two or more y’s, it is not a function.
3. If unsure, solve the defining equation for y – if it is not unique (e.g. involves a ± or even root), it fails.
2
Example 1 – Function test Is R = {(x, y) | x√ + y 2 = 4} a function?
2
Solution. Take x = 1: 1 + y = 4 ⇒ y = ± 3. Two y-values for one x, so R is not a function.
Example 2 – Domain, one-to-one, invertibility (tricky: restriction) Let f (x) = sin x on R.
Solution.
Df = R, image = [−1, 1], so f maps R onto [−1, 1] but not onto R.
f is not one-to-one: sin x = 0 has infinitely many solutions.
Hence f is not invertible. But the restriction g = f |[−π/2,π/2] is one-to-one and onto [−1, 1], so g is invertible,
with inverse sin−1 .
Lesson: a non-invertible function can become invertible once you restrict its domain.
√ 2
Example 3 – Composition (finding √ domains) f (x) = x, g(x) = x . Find f ◦ g, g ◦ f and their domains. √
2 2
Solution. (f ◦ g)(x) = f (x ) = x = |x|, defined for all x ∈ R since g(x) ≥ 0 always makes sense inside .
√ √ √
(g ◦ f )(x) = g( x) = ( x)2 = x, but only defined for x ≥ 0 (since x needs x ≥ 0 first).
Lesson: f ◦ g ̸= g ◦ f in general – check both the formula and the domain.
1.4 Implications
P ⇒ Q: “if P then Q”; converse: Q ⇒ P (not automatically true!)
Contrapositive: ¬Q ⇒ ¬P – always logically equivalent to P ⇒ Q.
P ⇔ Q: P and Q are equivalent (both P ⇒ Q and Q ⇒ P hold).
Negation of “f (x) has property A for all x ∈ U ” is “there exists x ∈ U such that f (x) does not have property
A”.
Counter-example: one single case where P fails disproves “P for all x”.
3
Calculus in Higher Dimensions
Complete Study Notes & Formula Sheets
Formula sheets · step-by-step methods · worked examples
for every section of Units 1–19
Part 1 – Basic Concepts: Units 1–3
Part 2 – Differentiation: Units 4–11
Part 3 – Integration: Units 12–19
Compiled as a revision companion to the UNISA MAT2615 study guides.
,MAT2615 Study Notes Calculus in Higher Dimensions
Contents
1 Mathematical Preliminaries 2
1.1 1.2–1.3 Sets, Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 1.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Concepts and Operations in n-Dimensional Euclidean Space 4
n
2.1 2.3–2.9 Vector Algebra in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 3
2.2 2.10 The Cross Product in R (only defined in R !) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
n
2.3 2.11 Lines in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
n
2.4 2.12 Planes in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
n
2.5 2.13 Subsets of R : Open, Closed, Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Functions 7
n m
3.1 3.1 Types of R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 3.2 Visualizing Functions: Graphs and Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
n p p m
3.3 3.3 Composition of R → R and R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Limits and Continuity 9
n
4.1 4.2–4.4 Limits of R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 4.5 Limits Along Curves – Proving a Limit Does Not Exist . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 4.6–4.7 Limits of Vector-Valued Functions & Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 The Derivative of a Real-Valued Function of One Variable 11
6 The Derivative of a Vector-Valued Function of One Variable 12
6.1 6.1–6.3 Vector Derivative, Chain Rule, Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . 12
6.2 6.4 Piecewise Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Derivatives of a Real-Valued Function of Several Variables 13
7.1 7.2–7.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.2 7.4–7.5 The Gradient and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.3 7.6 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.4 7.7 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.5 7.8 Tangent Planes to Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.6 7.9 Potential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.7 7.10 Higher Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
8 Derivatives of a Vector Field 17
8.1 8.2 The Derivative Matrix (Jacobian) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.2 8.3–8.4 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8.3 8.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
9 Taylor Polynomials 19
9.1 9.2 Taylor Polynomials for R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2 9.3 Taylor Polynomials for Rn → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10 Optimization 20
2
10.1 10.2 Local Extrema of R → R Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1
,MAT2615 Study Notes Calculus in Higher Dimensions
10.2 10.3 Constrained Optimization – Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
11 The Implicit Function Theorem 22
11.1 11.2 IFT for an Equation in 2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.2 11.3 IFT for an Equation in n Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.3 11.4–11.5 The Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
12 Single Integrals 24
12.1 12.2–12.4 Definite Integrals, FTC, Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
13 Double Integrals 25
13.1 13.5 Iterated Integrals over Type 1 and Type 2 Regions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
13.2 13.6 Change of Variables (Jacobian) and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 25
14 Triple Integrals 27
14.1 14.4–14.5 Iterated Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
14.2 14.6 Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
15 Line Integrals of Real-Valued Functions 29
15.1 15.2–15.3 Line Integrals w.r.t. Arc Length, x, and y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
15.2 15.5 Line Integrals over Piecewise Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
16 Line Integrals of Vector Fields 30
16.1 16.2 The Line Integral of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
16.2 16.5–16.7 The Fundamental Theorem of Line Integrals & Path Independence . . . . . . . . . . . . . . 30
17 Surface Integrals 32
17.1 17.2–17.4 Surface Integrals of Scalar Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
18 Flux Integrals 33
18.1 18.2–18.4 Setting Up and Evaluating Flux Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
19 The Theorems of Green, Gauss and Stokes 34
19.1 19.1 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
19.2 19.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
19.3 19.3 Gauss’s Theorem (The Divergence Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2
, MAT2615 Study Notes Calculus in Higher Dimensions
Unit 1 Mathematical Preliminaries
1.2–1.3 Sets, Relations and Functions
x ∈ S: x is an element of S x ∈ / S: x is not an element of S
A ∪ B = {x | x ∈ A or x ∈ B} A ∩ B = {x | x ∈ A and x ∈ B}
B − A = {x ∈ B | x ∈ / A} (complement of A relative to B)
A × B = {(a, b) | a ∈ A, b ∈ B} (product set); R × R = R2 , R × R × R = R3
Function: f : A → B such that every a ∈ A has exactly one image f (a) = b ∈ B.
Domain Df : set of inputs. Image/range f (A) = {f (a) | a ∈ A}.
Onto (surjective): every b ∈ B is hit at least once.
One-to-one (injective): f (x) = f (y) =⇒ x = y.
Invertible ⇐⇒ one-to-one and onto; inverse g satisfies g ◦ f = IA , f ◦ g = IB .
Composition: (f ◦ g)(x) = f (g(x)), domain = all x ∈ Dg with g(x) ∈ Df .
1. Write the relation as pairs (x, y) satisfying some condition.
2. Ask: for a given x, is there exactly one y? If some x gives two or more y’s, it is not a function.
3. If unsure, solve the defining equation for y – if it is not unique (e.g. involves a ± or even root), it fails.
2
Example 1 – Function test Is R = {(x, y) | x√ + y 2 = 4} a function?
2
Solution. Take x = 1: 1 + y = 4 ⇒ y = ± 3. Two y-values for one x, so R is not a function.
Example 2 – Domain, one-to-one, invertibility (tricky: restriction) Let f (x) = sin x on R.
Solution.
Df = R, image = [−1, 1], so f maps R onto [−1, 1] but not onto R.
f is not one-to-one: sin x = 0 has infinitely many solutions.
Hence f is not invertible. But the restriction g = f |[−π/2,π/2] is one-to-one and onto [−1, 1], so g is invertible,
with inverse sin−1 .
Lesson: a non-invertible function can become invertible once you restrict its domain.
√ 2
Example 3 – Composition (finding √ domains) f (x) = x, g(x) = x . Find f ◦ g, g ◦ f and their domains. √
2 2
Solution. (f ◦ g)(x) = f (x ) = x = |x|, defined for all x ∈ R since g(x) ≥ 0 always makes sense inside .
√ √ √
(g ◦ f )(x) = g( x) = ( x)2 = x, but only defined for x ≥ 0 (since x needs x ≥ 0 first).
Lesson: f ◦ g ̸= g ◦ f in general – check both the formula and the domain.
1.4 Implications
P ⇒ Q: “if P then Q”; converse: Q ⇒ P (not automatically true!)
Contrapositive: ¬Q ⇒ ¬P – always logically equivalent to P ⇒ Q.
P ⇔ Q: P and Q are equivalent (both P ⇒ Q and Q ⇒ P hold).
Negation of “f (x) has property A for all x ∈ U ” is “there exists x ∈ U such that f (x) does not have property
A”.
Counter-example: one single case where P fails disproves “P for all x”.
3
