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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 $5u_x+2u_y=0$.; ∇u is the gradient vector ($u_x$,$u_y$).

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 $5u_x+2u_y=0$.; ∇u is the gradient vector ($u_x$,$u_y$).

To expand (5,2)·∇u, substitute ∇$u = (u_x, u_y)$ and apply the definition of the Euclidean dot product: multiply corresponding components and add them. This yields $5u_x + 2u_y$.

Conditions: ∇$u = (u_x, u_y)$.; The dot product is defined as the sum of the products of corresponding components.

To expand (5,2)·∇u, substitute ∇$u = (u_x, u_y)$ and apply the definition of the Euclidean dot product: multiply corresponding components and add them. This yields $5u_x + 2u_y$.

Conditions: ∇$u = (u_x, u_y)$.; The dot product is defined as the sum of the products of corresponding components.

The gradient ∇u is the vector whose components are the first partial derivatives of u. For a function of two variables, ∇$u = (u_x, u_y)$, where $u_x$ is the partial derivative with respect to x and $u_y$ is the partial derivative with respect to y.

Conditions: u is a scalar function of two variables x and y.; u has partial derivatives with respect to x and y.

The gradient ∇u is the vector whose components are the first partial derivatives of u. For a function of two variables, ∇$u = (u_x, u_y)$, where $u_x$ is the partial derivative with respect to x and $u_y$ is the partial derivative with respect to y.

Conditions: u is a scalar function of two variables x and y.; u has partial derivatives with respect to x and y.