maths.free › Multivariable Calculus › Vector-Valued Functions of One Variable › Classic Calculus Approach

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

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.

Nígbà yìí Kò ní kọ́ǹpútà alágbèéká kan tí o lè fi àwọn à

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Àwọn Àtòjọ-ẹ̀yàn

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.

Àwọn ààyè ojú-ìwé yìí tí a fi pamọ́ láti Boelkins et al., Active Calculus Multivariable (CC BY-SA 4.0). Tí a fi pamọ́ síì níbẹ̀; àwọn àwọn àṣiṣe ní wa.

Diẹ̀ nínú Multivariable Calculus

Functions of several variables Partial derivatives and the gradient Optimisation in several variables Double and triple integrals Vector fields, line integrals and the big theorems