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Classic Calculus Approach

In this introduction, we will very briefly review some ideas and themes that ran through your work on single variable calculus because we will repeatedly use this same progression of ideas throughout our work in…

Classic Calculus Approach

In this introduction, we will very briefly review some ideas and themes that ran through your work on single variable calculus because we will repeatedly use this same progression of ideas throughout our work in multivariable calculus.

The key ideas from your early calculus courses included limits, derivatives, and integrals, which were used these to measure change and accumulation of scalar functions. Each of these ideas and associated applications were likely introduced as a way to measure a specific feature where you approximate the feature of interest, then looked at how to improve this approximation. Limits offered a powerful tool to precisely describe how (and when) these approximations converge to the measurement of interest. We will call this process the classic calculus approach.

Before we start our work on the calculus of vector-valued functions, we will review the ideas limits, differentiation, and integration as well as look at how the Classic Calculus Approach was used in the definition of each of these concepts. This review is not meant to be comprehensive but is intended to ensure that all students are reminded of key concepts that will be vital for our development of calculus for new types of of functions. We will provide links to activities where you can review and explore the ideas from single-variable calculus.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Symbols used here

\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\nabla f
gradient (nabla, del)
Vector of partial derivatives; points uphill.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\iint,\ \oint
double / contour integral
Integral over a region of the plane; integral around a closed curve.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).

Questions people ask

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen — the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

သင့်ရဲ့ကိုယ်ပိုင်စမ်းသပ်

Parts of this page are adapted from Boelkins et al., Active Calculus Multivariable (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

ပိုပြီး Multivariable Calculus

Functions of several variablesPartial derivatives and the gradientOptimisation in several variablesDouble and triple integralsVector fields, line integrals and the big theorems