How do you expand the dot product (5,2)·∇u?
Conditions
- ∇.
- The dot product is defined as the sum of the products of corresponding components.
Reasoning, step by step
- Write ∇u as (, ).
- Form the dot product (5,2)·(, ).
- Multiply the first components: .
- Multiply the second components: .
- Add the results: .
Example
The lecturer explains that the dot product multiplies the first components and adds the second components multiplied by each other. The board shows (5,2)·∇ and the spoken expansion gives .
Common misconceptions
- Multiplying the vectors element-wise without summing.
- Confusing the dot product with the cross product.
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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.
Rewriting the PDE as a dot product with the gradient allows the equation to be interpreted as a statement about directional derivatives. Specifically, it shows that the directional derivative of u in the direction (5,2) is zero, which implies that u is constant along curves in that direction.
Conditions: The PDE is .; ∇u is the gradient vector (,).
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).
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