MATH1011 - Lecture 18

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Path and Surface Integrals

• Recall that parametric equations are in the form: $r(t)=(x(t),y(t),z(t)),t\in [a,b]$
• A curve has many parametric equations
  • i.e. you can travel along a curve at any speed
• Thus, we can also choose any interval for $t$

See slides for how to find parametric forms

Length of a Curve

• The length of the curve given by a parametric equation is: $L(C)={\int }_a^b\left|\dot{\mathbf{r}}(t)\right|dt$

Note that this yields the formula seen in Chapter 6 for curves $(x,f(x))|x\in [a,b]$ given by a function in ${\mathbb{R}}^2$

See: MATH1011 - Lecture 12 > Length of a Curve

Path-Length Function

• Let $C$ be given in parametric form by $\{\mathbf{r}(t)|a\le t\le b\}$
• Let $s(t)$ denote the length of the path $C$ from $\mathbf{r}(a)$ to $\mathbf{r}(t)$ for $t\in [a,b]$
  • Further we travel on $C$ away from $\mathbf{r}(a)$ → longer the path becomes
  • Hence $s(t)$ is increasing and continuous

$L(C)={\int }_a^b\frac{ds}{dt}dt=\underset{C}{\int }ds$

Path Integrals

Suppose that along path $C$, the function $f(x,y)$ gives for the point $(x,y)\in C$ the height of the cliff

The path integral ${\int }_{C}fds$ is area under the function $f(x,y)$ along path $C$

• The path integral of $f$ over $C$ is:
  ${\int }_{C}fds={\int }_a^{b}f(\mathbf{r}(t))|\dot{\mathbf{r}}(t)|dt$

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MATH1011 - Lecture 19
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