Methods of Applied Mathematics Lecture Notes

1.3. VECTOR FIELDS AND DIFFERENTIAL FORMS 35possible to find coordinates u and v near a given point (not the origin) sothat−y dx + x dy = du? (1.149)4. A differential form usually cannot be transformed into a constant differentialform, but there special circumstances when that can occur. Is itpossible to find coordinates u and v near a given point (not the origin) sothat−yx 2 + y 2 dx + xx 2 dy = du? (1.150)+ y2 5. Consider the vector fieldL = x(4 − x − y) ∂∂x + (x − 2)y ∂ ∂y . (1.151)Find its zeros. At each zero, find the linearization. For each linearization,find the eigenvalues. Use this information to sketch the vector field.6. Let h be a smooth function. Its gradient expressed with respect to Cartesianbasis vectors ∂/∂x and ∂/∂y is∇h = ∂h ∂∂x ∂x + ∂h ∂∂y ∂y . (1.152)Find the gradient ∇h expressed with respect to polar basis vectors ∂/∂rand ∂/∂θ.7. Let H be a smooth function. Its Hamiltonian vector field expressed withrespect to Cartesian basis vectors ∂/∂x and ∂/∂y isˇ∇H = ∂H∂y∂∂x − ∂H∂x∂∂y . (1.153)Find this same vector field expressed with respect to polar basis vectors∂/∂r and ∂/∂θ.8. Consider the vector fieldL = (1 + x 2 + y 2 )y ∂∂x − (1 + x2 + y 2 )x ∂ ∂y . (1.154)Find its linearization at 0. Show that there is no coordinate system near0 in which the vector field L is expressed by its linearization. Hint: Solvethe associated system of ordinary differential equations, both for L andfor its linearization. Find the period of a solution in both cases.9. Here is an example of a fixed point where the linear stability analysisgives an elliptic fixed point but changing to polar coordinates shows theunstable nature of the fixed point:dxdtdydt= −y + x(x 2 + y 2 ) (1.155)= x + y(x 2 + y 2 ). (1.156)