4. Extra Credit. (10 Points.) A "mapping" from the plane to the plane is a function whose inputs are points on the plane and outputs are also points on the plane. To avoid confusion, let's think of this as a function from one copy of "the plane" to another, different copy of "the plane." For the first copy, we'll use the coordinates u and v, while for the second, we'll use the usual x and y. One example of a mapping is: x(u,v)=u2v,y(u,v)=u+v3. So the input (u,v) in the first plane (the uv-plane) would give the output (u2v,u+v3) in the second plane (the xy-plane). In this example, we'd say the point (1,2) in the uv-plane "maps to" the point (122,1+23)= (2,9) in the xy-plane. Consider a generic mapping defined by x(u,v) and y(u,v). Let's say we're interested in how this mapping behaves at (u0,v0) (in the uv-plane). Because mappings can be very complex, let's use linear approximation to simplify things: instead of working with x(u,v) and y(u,v) directly, we work with their linear approximations (let's call them Lx(u,v) and Ly(u,v)) at the base point (u0,v0), the idea being that the linear approximation mapping: x(u,v)=Lx(u,v),y(u,v)=Ly(u,v), will be a good approximation for our mapping when (u,v) is close to the base point. (a) Describe what the horizontal line (u0,v0)+t(1,0) in the uv-plane maps to under the linear approximation mapping. What about the vertical line (u0,v0)+t(0,1) ? (b) Describe, in a word, the geometric object that the rectangle with corner (u0,v0) and side lengths u and v maps to (under the linear approximation mapping). Remark: We call this object the "image" of our rectangle under this mapping. (c) What is the area of the rectangle in Part (b)? What is the area of the object it maps to? (d) What is the area of the image of the rectangle if the mapping is x(u,v)=ucosv,y(u,v)=usinv, when u=0 ? (Think about this for yourself: why the restriction u=0 ?) (e) How do you think this relates to integration