 An incompressible flow has the velocity components of u is x cubed plus 2 times z squared, and w is y cubed minus 2 times y times z. I want to determine the most general form of the y component of the velocity vector, which would satisfy the conservation of mass. To begin with, I recognize that I have an incompressible flow. That means when I deploy my conservation of mass, I can use the simplification partial derivative of u with respect to x plus partial derivative of v with respect to y plus partial derivative of w with respect to z is zero. We can use that simplification because incompressibility means the density doesn't change. If the density doesn't change, then we don't have to consider the change in density with respect to time, the change in density with respect to x, nor y, nor z. We can take that form of the conservation of mass and plug in the derivative of this equation with respect to x and this equation with respect to z. The derivative of u with respect to x is just going to be 3x squared because the second term is treated as a constant. The derivative of w with respect to z is just going to be minus 2 times y because the y cubed term is treated as a constant. When I plug those into this relationship, I can solve for the partial derivative of v with respect to y at which point I just have 2 times y minus 3x squared and that's the derivative of v with respect to y. That is not the v. It is not the y component of velocity. To get to the y component of velocity, I have to integrate both sides. If I bring del y to the other side of the equation, I can write this as two integrals. The integral of the first term with respect to y is going to be just y squared because we're taking 2 divided by 2. The derivative of the second term with respect to y, this comes out, this comes out, we're treated as a constant, so we have 3 times x squared times y. So the answer to the question is v is y squared minus 3 times x squared times y plus, theoretically, we could have a function of x and z as well because those would appear as constants.