Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(D_{\mathbf{u}} f(x, y)\) exists, then \(D_{\mathbf{u}} f(x, y)=-D_{-\mathbf{u}} f(x, y)\)

Short answer

The statement is true. If the directional derivative of \( f \) in the direction of \( \mathbf{u} \) exists, then the directional derivative of \( f \) in the direction of \( -\mathbf{u} \) is indeed the negative of the original derivative.

Step-by-step solution

Understanding the statement

First, understand what the statement is saying. A directional derivative at a given point measures the rate at which the function changes if one starts at that point and travels in a specific direction. So, if we have a directional derivative in one direction, the assertion is that the directional derivative in the negative of that direction should be the negative of the original directional derivative.

Generate a proof

Let's use the definition of the directional derivative. The directional derivative of \( f \) at a point \( (x,y) \) along a vector \( \mathbf{u} \) is defined as \( D_{\mathbf{u}}f(x,y) = f_x(x,y)u_1 + f_y(x,y)u_2 \), where \( \mathbf{u} = (u_1,u_2) \). Similarly, the directional derivative along \( -\mathbf{u} \) would be \( D_{-\mathbf{u}}f(x,y) = f_x(x,y)(-u_1) + f_y(x,y)(-u_2) \) which simplifies to \( -f_x(x,y)u_1 - f_y(x,y)u_2 \) or \( -D_{\mathbf{u}} f(x,y) \). Hence, the statement is true.

Conclusion

We have proved that if a directional derivative along a direction vector exists, then the directional derivative along the negative of that vector is the negative of the initial directional derivative. Since no counterexample could be found, the statement is indeed true.