Problem Set 19

  1. Easy
    1. Determine the values of \(\alpha\) and \(\beta\) such that \[ \mathbf F(x,y,z) = (y+z, \alpha x+ z, x+\beta y) \] is a conservative vector field. Write down the potential function \(f: \mathbb R^3 \to \mathbb R\) so that \(\mathbf F = \nabla f\).
    2. Let \(C\) be the curve given parametrically by \[ \gamma(t) = (t\cos(t), t\sin(t), t^2), \qquad 0\leq t \leq \frac\pi 2 \] Compute the line integral \( \int_C \mathbf F \cdot d\mathbf x\) where \(\mathbf F\) is the your solution from part (1).

  2. Easy
    Compute the line integral of curve \(\left\{x^2+y^2=1, y\geq 0\right\}\) (oriented counter clockwise) through the vector field \(\mathbf F(x,y) = (2xy^2+1,2x^2y+2y)\).

  3. Easy
    1. Show that the vector field \(\mathbf F(x,y,z) = (y\cos(xy), x\cos(xy)+e^z, ye^z)\) is conservative and find its scalar potential.
    2. Consider the curve \(C\) given by the intersection of the sets \[ C = \left\{x^2+y^2+z^2=4\right\} \cap \left\{x=0\right\} \cap \left\{z\geq 0\right\} \] oriented clockwise. Determine \(\int_C \mathbf F \cdot d\mathbf x\) if \(\mathbf F\) is the vector field given in part (1).

  4. Easy
    Evaluate the following line integrals both directly and by using Green's Theorem.
    1. \(\displaystyle \oint_C (x+2y) dx + (x-2y) dy \) where \(C\) is given by the union of the images of the following two functions on \([0,1]\): \(f(x) = x^2\) and \(g(x) = x\), positively oriented with respect to the area the curves bound.
    2. \(\displaystyle \oint_C (3x-5y) dx + (x+6y) dy \) where \(C\) is the ellipse \(x^2/4 + y^2=1\) oriented counter-clockwise.

  5. Medium
    1. Let \(D \subseteq \mathbb R^2\) be a regular region and \(\partial D = C\) be a piece-wise smooth simple closed curve, oriented positively. If \(A(D)\) is the area of \(D\), show that \[ A(D) = \oint_C x dy = \oint_C -y dx = \oint_C \frac12( -ydx+xdy) . \]
    2. Consider the disk of radius \(r\), \(D_r=\left\{x^2+y^2\leq r\right\} \subseteq \mathbb R^2\). Use any of the formulae from part (1) to compute the area of this disk. [Of course, you already know what the result should be!]
    3. In this question we will show that artificially adding boundaries does not affect the line integral. Let \(L_r\) be any diameter of \(D\). Show that if we break \(D\) into two regions \(D = D_1 \cup D_2\) and give the boundary of \(D_1\) and \(D_2\) positive orientations, then for any \(C^1\) vector field \(\mathbf F(x,y)\) we have \[ \oint_{\partial D} \mathbf F \cdot d\mathbf x = \oint_{\partial D_1} \mathbf F \cdot d\mathbf x + \oint_{\partial D_2} \mathbf F\cdot d\mathbf x. \]
    4. Use part (1) to compute the area of the lemniscate \(x^4=x^2-y^2\). [Be careful, as the lemniscate's boundary is not a simple closed curve.]

  6. Medium
    Let \(C\) be a smooth curve in\(\mathbb R^3\) parameterized by a vector function \(\mathbf r: [a,b] \to \mathbb R^3\). Define the identity vector field \(\mathbf F(x,y,z) = (x, y, z)\)
    1. Show that \[ \int_C \mathbf{F} \cdot \mathrm d\mathbf{r} = \frac{1}{2} \left( \vert\vert \mathbf{r}(b)\vert \vert^2 - \vert \vert \mathbf{r}(a)\vert\vert^2 \right) .\]
    2. What can you conclude about \(\int_C \mathbf F \cdot \mathrm d\mathbf r\) when \(C\) is a closed curve?

  7. Bonus
    For this problem, we will always be working in \(\mathbb R^3\). Let \(\Omega^0(\mathbb R^3)\) be the set of \(C^1\) functions \(\mathbb R^3 \to \mathbb R\) and \(\Omega^1(\mathbb R^3)\) be the set of \(C^1\) vector fields \(\mathbb R^3 \to \mathbb R^3\).
    1. Show that \(\Omega^0(\mathbb R^3)\) and \(\Omega^1(\mathbb R^3)\) are both \(\mathbb R\)-vector spaces. [This should be a short proof!]
    2. Show that \(\text{grad}: \Omega^0 \to \Omega^1\) and \(\text{curl}: \Omega^1 \to \Omega^1\) are linear operators.
    3. Recall that if \(Z\) is the set of closed vector fields, then \(Z = \text{ker}(\text{curl})\) and if \(B\) is the set of exact vector fields, then \(B = \text{im}(\text{grad})\). Show that \(B \subseteq Z\).
    4. If \(S \subseteq \mathbb R^3\), we define the de Rham cohomology of \(S\) of degree one to be \[ H^1(S) = Z/B. \] Show that if \(S\) is the complement of the \(z\)-axis in \(\mathbb R^3\) then \(H^1(S)\) is not the trivial vector space. [Hint: It suffices to show that \(Z \neq B\).]

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