Why rewrite the PDE as (5,2)·∇?
Conditions
- The PDE is .
- ∇u is the gradient vector (,).
Reasoning, step by step
- Start with the PDE .
- Identify the coefficients 5 and 2 as components of a vector (5,2).
- Rewrite the left-hand side as the dot product (5,2)·∇u.
- Interpret the dot product as the directional derivative of u in the direction (5,2).
- Conclude that the directional derivative is zero, implying u is constant along curves in that direction.
Example
The lecturer states: 'the first step… is to rewrite it in a particular form. And this will allow us to use arguments involving directional derivatives.' The board shows 'Rewrite PDE as (5,2)·∇'.
Common misconceptions
- Thinking the rewrite changes the equation; it is just a notational change.
- Confusing the direction vector (5,2) with separate axis values instead of a single direction in the plane.
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Related questions
The equation (5,2)·∇ implies that the directional derivative of u in the direction of the vector (5,2) is zero. This means that the rate of change of u is zero when moving in that specific direction.
Conditions: The direction vector is (5,2).; ∇u is the gradient of u.
The general solution is , where f is an arbitrary differentiable function of one variable. This form arises because u is constant along the characteristic curves , so u depends only on the invariant .
Conditions: u is constant along the characteristic curves.; f is an arbitrary function of one variable.
The slope of the characteristic lines is . This is derived from the direction vector (5,2), where the change in y is 2 and the change in x is 5, so .
Conditions: The direction vector is (5,2).; The characteristic lines are straight lines in the xy-plane.
u is constant along curves in the direction (5,2) because the directional derivative of u in that direction is zero. If the rate of change of a function is zero in a specific direction, the function's value does not change as one moves along any curve tangent to that direction.
Conditions: (5,2)·∇.; The curves considered are tangent to the direction (5,2).
Integrate the ordinary differential equation with respect to x. This yields , where c is an arbitrary constant representing the y-intercept.
Conditions: The ODE is .; c is an arbitrary real constant.
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