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If $E_{1}=\frac{\partial}{\partial y}-kx\frac{\partial}{\partial w}$ and $E_{2}=cos\psi\frac{\partial}{\partial x} + sin \psi\frac{\partial}{\partial w}$ span tangent space of $M$, where $M$ is a two-dimensional submanifold of $R^{4}$, where $\{\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial v}, \frac{\partial}{\partial w}\}$ is a basis of $R^{4}$. How can we find vectors that span normal bundle to $M$?

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Solve for vectors that are orthogonal to $E_1$ and $E_2$ at a point, $P$, on $M$ -- you will find a two-dimensional basis -- call that vector space spanned by that basis $N_P$. Then, $\cup_{P\in M} N_P$ is the normal bundle.

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