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[2020-02-06 Thu 13:33]
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Session 62: Gradient Fields | Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare

1 Chalkboard

Figure 1: Recall FCT for line integral with gradient fields

Figure 1: Recall FCT for line integral with gradient fields

Figure 2: Meaning of conservative field

Figure 2: Meaning of conservative field

Figure 3: How to know if vector fields is a gradient field

Figure 3: How to know if vector fields is a gradient field

Figure 4: Test for gradient field

Figure 4: Test for gradient field

Figure 5: Example of non gradient field

Figure 5: Example of non gradient field

Figure 6: For which values of \(a\) \(\vb{F}\) is a gradient field?

Figure 6: For which values of aa \(\vb{F}\) is a gradient field?

Figure 7: Example answer

Figure 7: Example answer

2 How can we say that a given \(\vb{F}\) is a conservative field?

2.1 Front

How can we say that a given $\vb{F}$ is a conservative field?

Let \(\vb{F} = M \vu{i} + N \vu{j}\). And using curl

2.2 Back

  • \(\vb{F}\) is continuously differentiable in a region DD
  • M=fxM = f_x
  • N=fyN = f_y
  • In DD, \(\vb{F} = \grad{f}\) for some f(x,y)  ⟹  My=Nxf(x,y) \implies M_y = N_x
  • My=fxyM_y = f_{xy}
  • Nx=fyxN_x = f_{yx}

Using two-dimensional curl of \(\vb{F}\)

\(\text{curl} \vb{F} = \grad{f} = N_x - M_y = 0\)

If \(\text{curl} \vb{F} = 0\), then \(\vb{F}\) is a gradient field

3 Show that this vector field is not conservative

3.1 Front

Show that this vector field is not conservative

\({\displaystyle \vb{F} = \frac{-y \vu{i} + x \vu{j}}{x^2 + y^2}}\)

3.2 Back

You can not apply the conversion criteria, and check that \(\text{curl} \vb{F} = 0\) because \(\vb{F}\) is not defined at origin (0,0)(0,0)

If you check it, My=Nx=y2−x2(x2+y2)2{\displaystyle M_y = N_x = \frac{y^2 - x^2}{(x^2 + y^2)^2}}

To show that \(\vb{F}\) is not conservative, you need to compute \({\displaystyle \oint_C \frac{-y}{x^2 + y^2} \dd{x} + \frac{x}{x^2 + y^2} \dd{y}}\). Let CC unit circle for example (you can get any closed curve). In this case, is 2π≠02 \pi \neq 0

4 How can we show a exact differential?

4.1 Front

How can we show a exact differential?

For example, show that \((xe^x + y)\dd{x} + x \dd{y}\) is exact

4.2 Back

  • \(M \dd{x} + N \dd{y}\) is exact differential if and only if \(\vb{F}\) is a gradient field
  • To show \(\vb{F}\) is a gradient field, we must show that \(\vb{F}\) is continuously differentiable and My=NxM_y = N_x for all x,yx,y
  • \(\vb{F}\) is continuously differentiable for all x,yx,y by inspection
  • My=Nx=1M_y = N_x = 1
  • Show is exact

5 What is the equation of a two dimensional curl of \(\vb{F}\)?

5.1 Front

What is the equation of a two dimensional curl of $\vb{F}$?

5.2 Back

  • \(\vb{F} = M \vu{i} + N \vu{j}\)
  • \(\text{curl} \vb{F} = N_x - M_y\), is a scalar function