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Magazine Problem 2
In Exercises 1.1-1.6, sketch the given vector field \(\mathbf{F}\). To get started, it may be helpful to get an idea of the general direction of the arrows \(\mathbf{F}(x, y)\) at points on the \(x\) and \(y\) -axes and in each of the four quadrants. $$ \mathbf{F}(x, y)=(2 x, y) $$
Problem 3
In Exercises 1.1-1.6, sketch the given vector field \(\mathbf{F}\). To get started, it may be helpful to get an idea of the general direction of the arrows \(\mathbf{F}(x, y)\) at points on the \(x\) and \(y\) -axes and in each of the four quadrants. $$ \mathbf{F}(x, y)=(y,-x) $$
Problem 4
In Exercises 1.1-1.6, sketch the given vector field \(\mathbf{F}\). To get started, it may be helpful to get an idea of the general direction of the arrows \(\mathbf{F}(x, y)\) at points on the \(x\) and \(y\) -axes and in each of the four quadrants. $$ \mathbf{F}(x, y)=(-2 y, x) $$
Problem 5
In Exercises 1.1-1.6, sketch the given vector field \(\mathbf{F}\). To get started, it may be helpful to get an idea of the general direction of the arrows \(\mathbf{F}(x, y)\) at points on the \(x\) and \(y\) -axes and in each of the four quadrants. $$ \mathbf{F}(x, y)=(y, x) $$
Problem 6
In Exercises 1.1-1.6, sketch the given vector field \(\mathbf{F}\). To get started, it may be helpful to get an idea of the general direction of the arrows \(\mathbf{F}(x, y)\) at points on the \(x\) and \(y\) -axes and in each of the four quadrants. $$ \mathbf{F}(x, y)=(-x, y) $$
Problem 7
(a) Find a smooth vector field \(\mathbf{F}\) on \(\mathbb{R}^{2}\) such that, at each point \((x, y), \mathbf{F}(x, y)\) is a unit vector normal to the parabola of the form \(y=x^{2}+c\) that passes through that point. (b) Find a smooth vector field \(\mathbf{F}\) on \(\mathbb{R}^{2}\) such that, at each point \((x, y), \mathbf{F}(x, y)\) is a unit vector tangent to the parabola of the form \(y=x^{2}+c\) that passes through that point.
Problem 8
(a) Find a smooth vector field \(\mathbf{F}\) on \(\mathbb{R}^{2}\) such that \(\|\mathbf{F}(x, y)\|=\|(x, y)\|\) and, at each point \((x, y)\) other than the origin, \(\mathbf{F}(x, y)\) is a vector normal to the curve of the form \(x y=c\) that passes through that point. (b) Find a smooth vector field \(\mathbf{F}\) on \(\mathbb{R}^{2}\) such that \(\|\mathbf{F}(x, y)\|=\|(x, y)\|\) and, at each point \((x, y)\) other than the origin, \(\mathbf{F}(x, y)\) is a vector tangent to the curve of the form \(x y=c\) that passes through that point.
Problem 10
Let \(\mathbf{F}(x, y)=(y, x)\). (This vector field also appears in Exercise 1.5.) (a) If \(a\) is a real number, show that \(\alpha(t)=\left(a\left(e^{t}+e^{-t}\right), a\left(e^{t}-e^{-t}\right)\right)\) and \(\alpha(t)=\left(a\left(e^{t}-\right.\right.\) \(\left.\left.e^{-t}\right), a\left(e^{t}+e^{-t}\right)\right)\) are integral paths of \(\mathbf{F}\). (b) Describe the integral curves, and sketch a representative sample of them. Indicate the direction in which the curves are traversed by the integral paths. (Hint: To find a relationship between \(x\) and \(y\) on the integral paths, compute \(x(t)^{2}\) and \(y(t)^{2}\).)
