Let \(\displaystyle \gamma:[1,2]\to\R^3\), \(\displaystyle \gamma(t)=(\ln t,2t,t^2)\).

(a) Determine the length of \(\displaystyle \gamma\).

(b) Determine the line integral of the vector field \(\displaystyle f(x,y,z)=(x,y,z)\) along the curve \(\displaystyle \gamma\).

Difficulty: 3.

Let \(\displaystyle C\) be the geometric curve \(\displaystyle \{(x,y)|\ |x|+|y|=a\}\). \(\displaystyle \int_C xy\ \hbox{d}s=?\)

Difficulty: 3.

Let \(\displaystyle \gamma: [0,2]\to \R^2\), \(\displaystyle (t)\mapsto (t,t^2)\). Compute the \(\displaystyle \int_{\gamma} (-y,x)\ \hbox{d}g\) line integral where \(\displaystyle g\) is the identity function.

Difficulty: 3.

Let \(\displaystyle \gamma\) be the semicircle which is the right part of the circle centered at \(\displaystyle 0\) with radius \(\displaystyle a\) (i.e those points satsifying \(\displaystyle x\geq 0\)). \(\displaystyle \int_{\gamma} x\ \hbox{d}y=?\)

Difficulty: 3.

Let \(\displaystyle \gamma\) be the semicircle which is the upper part of the circle centered at \(\displaystyle 0\) with radius \(\displaystyle a\) (i.e those points satsifying \(\displaystyle y\geq 0\)). \(\displaystyle \int_{\gamma}x^2\ \hbox{d}s=?\)

Difficulty: 3.

Determine the line integral of the vector field \(\displaystyle \displaystyle\left(\frac{x}{1+y},\frac{y}{2+x}\right)\) along the parabola \(\displaystyle y=x^2\) segment between the points \(\displaystyle (-1,1)\) and \(\displaystyle (1,1)\).

Difficulty: 3.

\(\displaystyle a)\quad \int_0^2 \sin x\ \hbox{d}\{x\}=?\qquad\qquad\qquad b)\quad \int_{\gamma} x^2\ \hbox{d}(y^2)=?\)

where \(\displaystyle \gamma\) is the triangle with vertices \(\displaystyle (0,0), (2,0), (0,1)\).

Difficulty: 4.

Calculate the line integral \(\displaystyle \int xy\;\dy\) on the curve in the figure.

Difficulty: 4.

Consider a map \(\displaystyle g:[a,b]\to\R\) as a one-dimensional curve. When is it rectifiable? What is its length?

Difficulty: 4.

Let \(\displaystyle g:[0,1]\to\R^2\) be a simple closed and rectifiable curve. Prove that

\(\displaystyle \int_{g}x^2\;\dx = \int_{g}e^{-\cos y^2}\;\dy = 0. \)

Difficulty: 4.

Let \(\displaystyle *:\R^p\times \R^q\to\R^r\) be bilinear, \(\displaystyle f:\R^q\to\R^p\) continuous and \(\displaystyle g:[a,b]\to\R^q\) a continuous curve. Show that

(a) if \(\displaystyle g\) is rectifiable, then \(\displaystyle \int_g f(\mathbf{x})*\mathrm{d}\mathbf{x}\) exists;

(b) if \(\displaystyle g\) is continuously differentiable, then \(\displaystyle \int_g f(\mathbf{x})*\mathrm{d}\mathbf{x}=\int_a^b f(g(t))*g'(t)\dt.\)

Difficulty: 4.