Day 30 - Calculus of Variations#

Announcements#

• Midterm 2 is posted

• Look for feedback from DC on projects

Seminars this Week#

Seminars this Week#

Clicker Question 30-1#

The generic segment, $ds$, of a curve in 2D Cartesian coordinates is given by

The integral of $ds$ from $s_1$ to $s_2$ gives the length of the curve, $l$. What is the correct expression for $l$?

1. $l={\int }_{s_1}^{s_2}ds$

2. $l={\int }_{s_1}^{s_2}\sqrt{(dx{)}^2+(dy{)}^2}$

3. $l={\int }_{s_1}^{s_2}\sqrt{1+(dy/dx{)}^2}dx$

4. $l={\int }_{s_1}^{s_2}\sqrt{(dx/dy{)}^2+1}dy$

5. More than one of the above

Clicker Question 30-2#

I can explain why:

where $Y(x)=y(x)+\alpha \eta (x)$, the true path plus an error term.

1. Yes, I can explain why

2. I think I can explain why

3. I’m having trouble seeing why

4. I don’t think I can explain why

Clicker Question 30-3#

For the function $Y(x)=y(x)+\alpha \eta (x)$, where $y(x)$ is the true path, $\eta (x)$ is a small error term, and $\alpha$ is a small parameter, what is the derivative of $Y(x)$ with respect to $\alpha$?

1. $y(x)$

2. $\eta (x)$

3. ${\eta }^{\prime }(x)$

4. $\alpha \eta (x)$

5. $y^{\prime }(x)+\alpha {\eta }^{\prime }(x)$

Clicker Question 30-4#

For the function $Y^{\prime }(x)=y^{\prime }(x)+\alpha {\eta }^{\prime }(x)$, what is the derivative of $Y^{\prime }(x)$ with respect to $\alpha$?

1. $y^{\prime }(x)$

2. ${\eta }^{\prime }(x)$

3. ${\eta }^{″}(x)$

4. $\alpha {\eta }^{\prime }(x)$

5. $y^{″}(x)+\alpha {\eta }^{″}(x)$

Clicker Question 30-5#

The “surface term” that we computed for ${\int }_{s_1}^{s_2}{\eta }^{\prime }(x)\frac{df}{d{y}^{\prime }}dx$ is:

I can explain why this surface term is equal to zero:

1. Yes, I can explain why

2. I think I can explain why

3. I’m having trouble seeing why

4. I don’t think I can explain why

5. I don’t know what a surface term is

Clicker Question 30-6#

We completed this derivation with the following mathematical statement:

where $\eta (x)$ is an arbitrary function. What does this imply about the term in square brackets?

1. The term in square brackets must be a pure function of $x$.

2. The term in square brackets must be a pure function of $y$.

3. The term in square brackets must be a pure function of $y^{\prime }$.

4. The term in square brackets must be zero.

5. The term in square brackets must be a non-zero constant.