Multivariable Calculus – Part A
Lecture 1
Tom Hebdige
10/2/2025
Tom Hebdige Multivariable Calculus – Part A 10/2/2025 1 / 20
About me
Tom Hebdige
Email: tom.hebdige@york.ac.uk
Research: Quantum Information and Computing
I quantum random number generators
I quantum thermodynamics
Tom Hebdige Multivariable Calculus – Part A 10/2/2025 2 / 20
Module Outline - Part A
The module is divided into two parts. In Part A we cover:
1 Functions of multiple variables
2 Taylor series
3 Fourier series
4 Double integration
5 Extrema
Part B runs in parallel and is almost entirely separate.
Tom Hebdige Multivariable Calculus – Part A 10/2/2025 3 / 20
Contact hours
Lectures
I 2 hours (out of 4) per week on Part A
I Recordings available on Panopto
I Please ask questions!
Seminars
I Weeks 2, 4, 6, 8, 10
I Feedback from assignments
I Practice questions in small groups
Computer practicals
I Weeks 3, 5, 7, 9, 11
I Using Maple to solve problems relating to material
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Course materials
Lecture notes
I Provided in both pdf and html formats
I Supplement with your own notes!
Slides and visualiser notes
Discussion forum on Blackboard
Textbooks
I Recommended text: “Thomas’ Calculus (15th edition with SI units)”
by Hass, Heil, Bogacki & Weir
I Digital copy available through Kortext
Problem sets
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Assessment
Assignments (10%)
I 4 assignments, best 3 marks will be taken
I 2 from Part A, 2 from Part B
I Extensions to the assignments are not possible
I Submission on Blackboard as pdf
I Feedback in seminars
Computer quizzes (10%)
I 4 quizzes, best 3 marks will be taken
I Based on computer practicals
I Multiple choice quizzes on Blackboard
Exam (80%)
I 2 hours
I 50% of marks from each part
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How to succeed in this course
George Polya in ‘How to Solve It’:
Mathematics, you see, is not a spectator sport. To understand
mathematics means to be able to do mathematics. And what
does it mean [to be] doing mathematics? In the first place, it
means to be able to solve mathematical problems.
Do the problem sheets!
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Plan for today
1 Functions of multiple variables
2 Partial derivatives
3 Generalised chain rules
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Graph of a function
p
Figure: Surface defined by f (x, y ) = 1 − x 2 − y 2 on the domain
D = {(x, y ) ∈ R2 | x 2 + y 2 ≤ 1}.
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Level sets
Figure: Contour plot for f (x, y ) = x 2 + y 2 with contours representing values
f (x, y ) = 0, 10, 20, 30, 40, 50.
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Contour map of local area
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Functions of multiple variables
A real-valued function of n variables with domain D ⊆ Rn is
f : D → R
(x1 , · · · , xn ) → f (x1 , · · · , xn )
If f is a function of two variables, then its graph is the surface
{(x, y , z) ∈ R3 | z = f (x, y )} .
If f is a function of two variables, then a level set for f is the set
{(x, y ) ∈ R2 | f (x, y ) = c}
for some constant c ∈ R.
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Partial derivatives – notations
Unfortunately partial derivatives have a variety of notations:
∂f
≡ ∂x f (x, y ) ≡ fx (x, y ) ≡ f1 (x, y )
∂x
∂f
≡ ∂y f (x, y ) ≡ fy (x, y ) ≡ f2 (x, y ) .
∂y
∂f
I will largely stick to the ∂x notation.
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Partial derivatives – notations
Alternative notations for second order partial derivatives:
∂2f
∂ ∂f
= = fxx = f11 (x, y )
∂x ∂x ∂x 2
∂2f
∂ ∂f
= = fyy = f22 (x, y )
∂y ∂y ∂y 2
∂2f
∂ ∂f
= = fxy = (fx )y = f12 (x, y )
∂y ∂x ∂y ∂x
∂2f
∂ ∂f
= = fyx = (fy )x = f21 (x, y )
∂x ∂y ∂x∂y
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Partial derivatives
For a function f (x, y ):
the partial derivative of f with respect to x is
∂f f (x + h, y ) − f (x, y )
:= lim
∂x h→0 h
the partial derivative of f with respect to y is
∂f f (x, y + h) − f (x, y )
:= lim
∂y h→0 h
Second order derivatives:
∂2f ∂2f
∂ ∂f ∂ ∂f
= =
∂x ∂x ∂x 2 ∂y ∂y ∂y 2
∂2f ∂2f
∂ ∂f ∂ ∂f
= =
∂y ∂x ∂y ∂x ∂x ∂y ∂x∂y
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Clairaut’s theorem
For sufficiently smooth functions,
∂2f ∂2f
=
∂x∂y ∂y ∂x
i.e. order of partial differentiation doesn’t matter.
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Generalised chain rule
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Generalised chain rule
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Generalised chain rule
1 independent variable, 2 intermediate variables:
df dx ∂f dy ∂f
= +
dt dt ∂x dt ∂y
2 independent variables, 3 intermediate variables:
∂f ∂f ∂x ∂f ∂y ∂f ∂z
= + +
∂r ∂x ∂r ∂y ∂r ∂z ∂r
∂f ∂f ∂x ∂f ∂y ∂f ∂s
= + +
∂s ∂x ∂s ∂y ∂s ∂z ∂r
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Gradient
Gradient
∂f ∂f
∇f := i+ j
∂x ∂y
Chain rule as dot product:
dF dr
(t) = (t) · ∇f [r(t)]
dt dt
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