91Ó°ÊÓ

In Problems 1-16, evaluate the given integral along the indicated contour. \(\int_{C}(z+3) d z\), where \(C\) is \(x=2 t, y=4 t-1,1 \leq t \leq 3\)

In Problems 1-8, show that \(\oint_{C} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\) $$ f(z)=z^{3}-1+3 i $$

In Problems 1-4, for the given velocity field \(\mathbf{F}(x, y)\), verify that \(\operatorname{div} \mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) in an appropriate domain \(D\). $$ \mathbf{F}(x, y)=\left(\cos \theta_{0}\right) \mathbf{i}+\left(\sin \theta_{0}\right) \mathbf{j}, \theta_{0} \text { a constant } $$

In Problems 1-10, evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{-1}^{3} x(x-1)(x+2) d x $$

Evaluate the definite integral. If necessary, review the techniques of integration in your calculus text. $$ \int_{-1}^{0} t^{2} d t+\int_{0}^{2} x^{2} d x+\int_{2}^{3} u^{2} d u $$

For the given velocity field \(\mathbf{F}(x, y)\), verify that \(\operatorname{div} \mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) in an appropriate domain \(D\). $$ \mathbf{F}(x, y)=-y \mathbf{i}-x \mathbf{j} $$

Evaluate the given integral along the indicated contour. \(\int_{C}(2 \bar{z}-z) d z\), where \(C\) is \(x=-t, y=t^{2}+2,0 \leq t \leq 2\)

Evaluate the given integral along the indicated contour. \(\int_{C} z^{2} d z\), where \(C\) is \(z(t)=3 t+2 i t,-2 \leq t \leq 2\)

Show that \(\oint_{C} f(z) d z=0\), where \(f\) is the given function and \(C\) is the unit circle \(|z|=1\) $$ f(z)=\frac{z}{2 z+3} $$

For the given velocity field \(\mathbf{F}(x, y)\), verify that \(\operatorname{div} \mathbf{F}=0\) and curl \(\mathbf{F}=\mathbf{0}\) in an appropriate domain \(D\). $$ \mathbf{F}(x, y)=2 x \mathbf{i}+(3-2 y) \mathbf{j} $$