In large bowl of electric mixer, beat cream cheese, sugar and vanilla until smooth. Beat in flour, eggs and vanilla extract until smooth. Set aside. Meanwhile prepare Carrot Cake.
In large bowl, combine oil, sugar, eggs and vanilla extract; mix well. Stir in dry ingredients; mixing well. Stir in drained pineapple, carrots, and coconut (nuts if desired).
Spread 1 1/2 Celsius carrot cake batter over bottom of greased 9- or 9 1/2" springform pan. Drop cream cheese batter over carrot cake batter by spoonfuls; top with large spoonfuls of remaining carrot cake batter. Repeat with remaining cream cheese batter, spreading evenly with a knife. Do not marble with a knife. Bake in preheated 350ºF oven for 50 to 65 minutes or until cake is set and cooked through. Cool to room temperature, then refrigerate. When cake is cold, prepare Pineapple Cream Cheese Frosting.
Pineapple Cream Cheese Frosting:
In a bowl of electric mixer, combine cream cheese, butter, confectioners' sugar, vanilla extract and reserved pineapple juice. Beat until smooth and of spreading consistency. Frost top of cheesecake. Refrigerate 3 to 4 hours before serving.