Problem Set 18

  1. Easy
    Plot the following vector fields:
    1. \(F(x,y) = (\sqrt x, y)\),
    2. \(F(x,y) = (y,x)\),
    3. \(F(x,y) = \left( \frac1x, y\right)\),
    4. \(F(x,y) = \left(\frac yx, y\right)\),
    5. \(F(x,y) = (x+y,x-y)\).

  2. Easy
    For each of the following functions, compute the gradient, Laplacian, divergence, and curl, if it makes sense to do so.
    1. \(F(x,y) = \sin(xe^y)\),
    2. \(F(x,y) = x^2+y^2-2z^2\),
    3. \(F(x,y,z) = (\sin^2(x), \cos^2(z), xyz)\),
    4. \(F(x,y,z) = \frac{ x}{x^2+y^2}\),
    5. \(F(x,y) = \log(x^2+y^2)\),
    6. \(F(x,y,z) = (2xy^2z^2, 2yx^2z^2,2zx^2y^2)\),
    7. \(F(x,y,z,w) = (e^{zw}, e^{xz}, e^{xw}, e^{yz})\).

  3. Easy
    Show that the following identities hold:
    1. \(\nabla(fg) = f\nabla g + g\nabla f\)
    2. \(\text{curl}(f\mathbf G) = f \text{curl} \mathbf G + (\nabla f)\times \mathbf G\)
    3. \(\text{div}(f\mathbf G) = f \text{div} \mathbf G + (\nabla f) \cdot \mathbf G\)
    4. \(\text{div}(\mathbf F\times \mathbf G) = \mathbf G \cdot (\text{curl} \mathbf F) - \mathbf F \cdot (\text{curl}\mathbf G)\)

  4. Easy
    Show that the following identities hold:
    1. \(\text{curl}(\nabla f) = 0\)
    2. \(\text{div}(\text{curl} \mathbf F) = 0\)

  5. Easy
    Find the arclength of the following curves
    1. The straight line between \((1,2,3)\) and \((3,1,2)\),
    2. The curve given by \(y^2=x^3\) between \((1,1)\) and \((4,8)\),
    3. The curve given by \((x,y) = (t-\sin(t),1-\cos(t))\) for \(0 \leq t \leq 2\pi\),
    4. The curve given by \((x,y) = (\cos^3(t),\sin^3(t))\) for \(0\leq t\leq \frac\pi2\),
    5. The graph of a \(C^1\) function \(y=f(x)\) for \(a \leq x \leq b\).

  6. Easy
    Let \(\mathbf F(x,y) = (-y^2,xy)\) and \(C = \displaystyle\left\{ \frac{x^2}{a^2} + \frac{y^2}{b^2} =1: y \geq 0\right\}\). Determine \(\displaystyle \int_C \mathbf F \cdot d\mathbf x\) if \(C\) is oriented counter clockwise when viewed from \((0,0,1)\).

  7. Easy
    Determine \(\int_C \mathbf F \cdot dx\) where \(\mathbf F(x,y) = (x^2,-y)\) and \(C\) is the graph of \(y=e^x\) from \((2,e^2)\) to \((0,1)\).

  8. Easy
    Determine \(\int_C \mathbf F \cdot dx\) where \(\mathbf F(x,y,z) = (z,-y,x)\) and \(C\) is the line segment between the points \((5,0,2)\) and \((5,3,4)\).

  9. Easy
    Let \(\mathbf F(x,y,z) = (x,y,z^2)\) and \(C\) be the curve given by the intersection of the cylinder \(x^2+y^2=1\) and \(z=x\), with any orientation. Determine \(\oint_C \mathbf F \cdot dx\).

  10. Easy
    By explicitly parameterizing, show that if \(C\) is the constant curve (that is, the curve which consists of a single point), then for any vector field \(\mathbf F\) we have \[ \int_C \mathbf F \cdot dx = 0. \]

  11. Easy
    Consider a vertical line segment \(S=\left\{(x,y): a \leq y \leq b, x=c\right\}\) in \(\mathbb R^2\), for constants \(a,b,c\). Show that for any vector field \(\mathbf F(x,y) = (F_1(x,y), F_2(x,y)\) that the line integral does not depend on \(F_1\). Similarly conclude that for a horizontal line segment, the line integral does not depend on \(F_2\).

  12. Medium
    Let \((x,y)\) be the standard Cartesian coordinates in \(\mathbb R^2\). Changing to polar coordinates we set \[ x = r\cos(\theta), \quad y = r\sin(\theta). \]
    1. Let \(e_x = (1,0)\) and \(e_y = (0,1)\) be the standard Cartesian unit vectors of \(\mathbb R^2\). Let us define \(e_r=\cos(\theta) e_x + \sin(\theta) e_y\) and \(e_\theta=-\sin(\theta) e_x + \cos(\theta) e_y\), to be the standard polar unit ''vectors" (at the point \((r, \theta\))).
    2. Using the multivariable chain rule, determine the \(\nabla\) operator in polar coordinates.
    3. Compute the Laplacian \(\nabla^2\) in polar coordinates.

  13. Medium
    Let \(\mathbf F(x,y,z) = (yz, xz, xy)\) and define \[ C_{r,h} = \left\{ (x,y,z): x^2+y^2=r^2, z = h\right\}. \] Show that for and \(r >0\) and \(h \in \mathbb R\), \[ \int_{C_{r,h}} \mathbf F \cdot dx = 0. \]

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